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Ft(B)5 b(;)17 b(C)7 b Fx(\))27 b(:)849 4794 y Ft(g)t(f)37 b Fx(=)28 b Ft(hf)43 b Fx(=)-17 b Fu(\))33 b Ft(g)e Fx(=)c Ft(h)98 b Fx(\()p Ft(f)43 b Fx(is)32 b(righ)m(t)g(cancellable\))n Ft(:)149 4983 y Fv(De\014nition)37 b(8.1.8.)49 b Fx(Giv)m(en)40 b Ft(A;)17 b(B)45 b Fu(2)40 b(C)6 b Fx(.)65 b(An)41 b(ob)5 b(ject)40 b Ft(A)27 b Fu(\002)h Ft(B)45 b Fx(in)39 b Fu(C)46 b Fx(together)40 b(with)f(mor-)0 5099 y(phisms)f Ft(p)381 5114 y Fq(A)476 5099 y Fx(:)g Ft(A)27 b Fu(\002)f Ft(B)43 b Fu(\000)-60 b(!)38 b Ft(A)h Fx(and)g Ft(p)1373 5114 y Fq(B)1471 5099 y Fx(:)f Ft(A)27 b Fu(\002)f Ft(B)44 b Fu(\000)-60 b(!)38 b Ft(B)43 b Fx(is)38 b(called)g(a)g(\(categorical\))f Fs(pr)-5 b(o)g(duct)38 b Fx(of)0 5216 y Ft(A)k Fx(and)f Ft(B)47 b Fx(if)40 b(for)i(ev)m(ery)h(ob)5 b(ject)42 b Ft(T)57 b Fu(2)44 b(C)j Fx(and)42 b(ev)m(ery)i(pair)c(of)h(morphisms) f Ft(f)54 b Fx(:)43 b Ft(T)57 b Fu(\000)-60 b(!)43 b Ft(A)f Fx(and)p eop %%Page: 3 3 3 2 bop 1488 236 a Fo(1.)77 b(CA)-6 b(TEGORIES)1424 b(3)0 425 y Ft(g)31 b Fx(:)d Ft(T)41 b Fu(\000)-60 b(!)28 b Ft(B)33 b Fx(there)c(exists)g(a)f(unique)h(morphism)d(\()p Ft(f)5 b(;)17 b(g)t Fx(\))27 b(:)h Ft(T)41 b Fu(\000)-60 b(!)28 b Ft(A)13 b Fu(\002)g Ft(B)34 b Fx(suc)m(h)c(that)e(the)g (diagram)1763 566 y Ft(T)1453 793 y Fq(f)1647 684 y Fl(\000)1564 767 y(\000)1481 850 y(\000)1398 933 y(\000)1379 951 y(\000)-83 b(\011)2103 784 y Fq(g)1867 684 y Fl(@)1950 767 y(@)2033 850 y(@)2116 933 y(@)2135 951 y(@)g(R)1275 1050 y Ft(A)314 b(A)22 b Fu(\002)g Ft(B)p 1376 1022 257 4 v 1376 1020 a Fl(\033)1461 988 y Fq(p)1497 999 y Fk(A)2246 1050 y Ft(B)p 1964 1022 254 4 v 2135 1020 a Fl(-)2047 988 y Fq(p)2083 999 y Fk(B)p 1797 951 4 351 v 1799 951 a Fl(?)1838 793 y Fn(\()p Fq(f)s(;g)r Fn(\))0 1177 y Fx(comm)m(utes.)149 1293 y(An)39 b(ob)5 b(ject)38 b Ft(E)43 b Fu(2)37 b(C)44 b Fx(is)37 b(called)g(a)g Fs(\014nal)i(obje)-5 b(ct)38 b Fx(if)e(for)i(ev)m(ery)h(ob)5 b(ject)39 b Ft(T)50 b Fu(2)37 b(C)44 b Fx(there)39 b(exists)f(a)0 1409 y(unique)33 b(morphism)e Ft(e)d Fx(:)f Ft(T)42 b Fu(\000)-60 b(!)28 b Ft(E)38 b Fx(\(i.e.)43 b(Mor)1636 1424 y Fr(C)1681 1409 y Fx(\()p Ft(T)8 b(;)17 b(E)6 b Fx(\))33 b(consists)g(of)f (exactly)h(one)g(elemen)m(t\).)149 1525 y(A)h(category)f Fu(C)40 b Fx(whic)m(h)34 b(has)f(a)g(pro)s(duct)h(for)e(an)m(y)i(t)m(w) m(o)g(ob)5 b(jects)35 b Ft(A)e Fx(and)g Ft(B)39 b Fx(and)33 b(whic)m(h)h(has)f(a)0 1642 y(\014nal)f(ob)5 b(ject)33 b(is)f(called)g(a)g(category)h(with)f(\014nite)g(pro)s(ducts.)149 1802 y Fv(Remark)37 b(8.1.9.)50 b Fx(If)27 b(the)h(pro)s(duct)g(\()p Ft(A)12 b Fu(\002)g Ft(B)5 b(;)17 b(p)1885 1817 y Fq(A)1943 1802 y Ft(;)g(p)2036 1817 y Fq(B)2096 1802 y Fx(\))28 b(of)f(t)m(w)m(o)h(ob)5 b(jects)29 b Ft(A)f Fx(and)g Ft(B)k Fx(in)27 b Fu(C)34 b Fx(exists)0 1918 y(then)f(it)f(is)g(unique) h(up)g(to)f(isomorphism.)149 2034 y(If)h(the)g(\014nal)f(ob)5 b(ject)33 b Ft(E)39 b Fx(in)31 b Fu(C)39 b Fx(exists)34 b(then)f(it)e(is)h(unique)h(up)g(to)g(isomorphism.)149 2194 y Fv(Problem)j(8.1.1.)49 b Fx(Let)33 b Fu(C)38 b Fx(b)s(e)33 b(a)f(category)h(with)f(\014nite)g(pro)s(ducts.)44 b(Giv)m(e)32 b(a)g(de\014nition)f(of)h(a)0 2310 y(pro)s(duct)g(of)f(a)g (family)e Ft(A)925 2325 y Fn(1)964 2310 y Ft(;)17 b(:)g(:)g(:)33 b(;)17 b(A)1273 2325 y Fq(n)1351 2310 y Fx(\()p Ft(n)27 b Fu(\025)i Fx(0\).)42 b(Sho)m(w)33 b(that)e(pro)s(ducts)h(of)f(suc)m (h)i(families)28 b(exist)k(in)0 2427 y Fu(C)6 b Fx(.)149 2587 y Fv(De\014nition)37 b(and)h(Remark)f(8.1.10.)49 b Fx(Let)39 b Fu(C)45 b Fx(b)s(e)40 b(a)e(category)-8 b(.)63 b(Then)40 b Fu(C)2946 2551 y Fq(op)3059 2587 y Fx(with)e(the)i(fol-)0 2703 y(lo)m(wing)28 b(data)g(Ob)17 b Fu(C)726 2667 y Fq(op)828 2703 y Fx(:=)27 b(Ob)17 b Fu(C)6 b Fx(,)30 b(Mor)1396 2718 y Fr(C)1437 2699 y Fk(op)1508 2703 y Fx(\()p Ft(A;)17 b(B)5 b Fx(\))27 b(:=)h(Mor)2114 2718 y Fr(C)2159 2703 y Fx(\()p Ft(B)5 b(;)17 b(A)p Fx(\),)30 b(and)f Ft(f)d Fu(\016)2798 2718 y Fq(op)2886 2703 y Ft(g)31 b Fx(:=)d Ft(g)18 b Fu(\016)d Ft(f)39 b Fx(de\014nes)0 2819 y(a)32 b(new)i(category)-8 b(,)33 b(the)g Fs(dual)h(c)-5 b(ate)g(gory)33 b Fx(to)f Fu(C)6 b Fx(.)149 2979 y Fv(Remark)37 b(8.1.11.)50 b Fx(An)m(y)35 b(notion)e(expressed)38 b(in)33 b(categorical)g(terms)h(\(with)g(ob)5 b(jects,)36 b(mor-)0 3096 y(phisms,)j(and)e(their)g(comp)s(osition\))f(has)i(a)f Fs(dual)i(notion)p Fx(,)g(i.e.)58 b(the)38 b(giv)m(en)f(notion)g(in)g (the)h(dual)0 3212 y(category)-8 b(.)149 3328 y(Monomorphisms)31 b Ft(f)42 b Fx(in)31 b(the)h(dual)f(category)h Fu(C)1920 3292 y Fq(op)2026 3328 y Fx(are)g(epimorphisms)e(in)h(the)h(original)c (cat-)0 3444 y(egory)33 b Fu(C)39 b Fx(and)33 b(con)m(v)m(ersely)-8 b(.)46 b(A)33 b(\014nal)e(ob)5 b(jects)34 b Ft(I)41 b Fx(in)32 b(the)h(dual)f(category)h Fu(C)2731 3408 y Fq(op)2838 3444 y Fx(is)f(an)h Fs(initial)h(obje)-5 b(ct)0 3561 y Fx(in)32 b(the)h(original)c(category)k Fu(C)6 b Fx(.)149 3721 y Fv(De\014nition)37 b(8.1.12.)49 b Fx(The)35 b Fs(c)-5 b(opr)g(o)g(duct)33 b Fx(of)g(t)m(w)m(o)i(ob)5 b(jects)35 b(in)e(the)h(category)g Fu(C)40 b Fx(is)33 b(de\014ned)i(to)0 3837 y(b)s(e)e(a)f(pro)s(duct)h(of)f(the)h(ob)5 b(jects)34 b(in)e(the)h(dual)e(category)i Fu(C)2142 3801 y Fq(op)2216 3837 y Fx(.)149 3997 y Fv(Remark)k(8.1.13.)50 b Fx(Equiv)-5 b(alen)m(t)41 b(to)g(the)g(preceding)h(de\014nition)e(is) h(the)h(follo)m(wing)c(de\014ni-)0 4113 y(tion.)149 4229 y(Giv)m(en)31 b Ft(A;)17 b(B)32 b Fu(2)c(C)6 b Fx(.)43 b(An)31 b(ob)5 b(ject)31 b Ft(A)18 b Fu(q)f Ft(B)35 b Fx(in)30 b Fu(C)36 b Fx(together)31 b(with)f(morphisms)f Ft(j)2953 4244 y Fq(A)3038 4229 y Fx(:)e Ft(A)h Fu(\000)-60 b(!)28 b Ft(A)18 b Fu(q)f Ft(B)0 4346 y Fx(and)29 b Ft(j)226 4361 y Fq(B)315 4346 y Fx(:)f Ft(B)33 b Fu(\000)-60 b(!)27 b Ft(A)15 b Fu(q)g Ft(B)33 b Fu(\000)-60 b(!)28 b Ft(B)34 b Fx(is)29 b(a)g(\(categorical\))e(copro)s(duct)j(of)f Ft(A)g Fx(and)g Ft(B)34 b Fx(if)28 b(for)h(ev)m(ery)i(ob)5 b(ject)0 4462 y Ft(T)42 b Fu(2)29 b(C)39 b Fx(and)33 b(ev)m(ery)i(pair)d(of)h(morphisms)e Ft(f)39 b Fx(:)29 b Ft(A)f Fu(\000)-60 b(!)29 b Ft(T)46 b Fx(and)34 b Ft(g)d Fx(:)e Ft(B)k Fu(\000)-60 b(!)29 b Ft(T)46 b Fx(there)34 b(exists)g(a)e(unique)0 4578 y(morphism)f([)p Ft(f)5 b(;)17 b(g)t Fx(])27 b(:)h Ft(A)22 b Fu(q)g Ft(B)33 b Fu(\000)-60 b(!)27 b Ft(T)47 b Fx(suc)m(h)34 b(that)e(the)h(diagram) 1763 5250 y Ft(T)1453 4988 y Fq(f)1379 4879 y Fl(@)1462 4962 y(@)1545 5045 y(@)1628 5128 y(@)1647 5147 y(@)-83 b(R)2103 4980 y Fq(g)2135 4879 y Fl(\000)2052 4962 y(\000)1969 5045 y(\000)1886 5128 y(\000)1867 5147 y(\000)g(\011)1275 4762 y Ft(A)315 b(A)22 b Fu(q)g Ft(B)p 1376 4730 258 4 v 1551 4728 a Fl(-)1466 4696 y Fq(j)1495 4707 y Fk(A)2246 4762 y Ft(B)p 1963 4730 255 4 v 1963 4728 a Fl(\033)2049 4696 y Fq(j)2078 4707 y Fk(B)p 1797 5147 4 351 v 1799 5147 a Fl(?)1838 4988 y Fn([)o Fq(f)s(;g)r Fn(])p eop %%Page: 4 4 4 3 bop 0 236 a Fo(4)1494 b(8.)77 b(TOOLBO)n(X)0 425 y Fx(comm)m(utes.)149 541 y(The)43 b(category)f Fu(C)48 b Fx(is)41 b(said)g(to)g(ha)m(v)m(e)i Fs(\014nite)g(c)-5 b(opr)g(o)g(ducts)41 b Fx(if)f Fu(C)2437 505 y Fq(op)2553 541 y Fx(is)h(a)g(category)h(with)f(\014nite)0 658 y(pro)s(ducts.)j(In) 33 b(particular)e(copro)s(ducts)i(are)g(unique)g(up)g(to)f (isomorphism.)p eop %%Page: 5 5 5 4 bop 1527 236 a Fo(2.)77 b(FUNCTORS)1463 b(5)1497 425 y Fv(2.)94 b(F)-9 b(unctors)149 600 y(De\014nition)37 b(8.2.1.)49 b Fx(Let)33 b Fu(C)38 b Fx(and)33 b Fu(D)i Fx(b)s(e)e(categories.)43 b(Let)33 b Fu(F)42 b Fx(consist)33 b(of)149 743 y(1.)42 b(a)32 b(map)g(Ob)17 b Fu(C)34 b(3)28 b Ft(A)f Fu(7!)h(F)10 b Fx(\()p Ft(A)p Fx(\))27 b Fu(2)h Fx(Ob)17 b Fu(D)s Ft(;)149 860 y Fx(2.)42 b(a)32 b(family)e(of)i(maps) 348 1031 y Fu(fF)470 1046 y Fq(A;B)631 1031 y Fx(:)27 b(Mor)861 1046 y Fr(C)906 1031 y Fx(\()p Ft(A;)17 b(B)5 b Fx(\))28 b Fu(3)g Ft(f)38 b Fu(7!)28 b(F)1586 1046 y Fq(A;B)1718 1031 y Fx(\()p Ft(f)11 b Fx(\))28 b Fu(2)g Fx(Mor)2151 1046 y Fr(D)2212 1031 y Fx(\()p Fu(F)10 b Fx(\()p Ft(A)p Fx(\))p Ft(;)17 b Fu(F)10 b Fx(\()p Ft(B)5 b Fx(\)\))p Fu(j)p Ft(A;)17 b(B)31 b Fu(2)d(C)6 b(g)262 1218 y Fx([or)32 b Fu(fF)530 1233 y Fq(A;B)690 1218 y Fx(:)c(Mor)921 1233 y Fr(C)966 1218 y Fx(\()p Ft(A;)17 b(B)5 b Fx(\))28 b Fu(3)g Ft(f)38 b Fu(7!)27 b(F)1645 1233 y Fq(A;B)1778 1218 y Fx(\()p Ft(f)11 b Fx(\))27 b Fu(2)h Fx(Mor)2211 1233 y Fr(D)2271 1218 y Fx(\()p Fu(F)10 b Fx(\()p Ft(B)5 b Fx(\))p Ft(;)17 b Fu(F)10 b Fx(\()p Ft(A)p Fx(\)\))p Fu(j)p Ft(A;)17 b(B)31 b Fu(2)d(C)6 b(g)p Fx(])0 1369 y Fu(F)42 b Fx(is)32 b(called)f(a)i Fs(c)-5 b(ovariant)32 b Fx([)p Fs(c)-5 b(ontr)g(avariant)9 b Fx(])32 b Fs(functor)h Fx(if)149 1513 y(1.)42 b Fu(F)339 1528 y Fq(A;A)468 1513 y Fx(\(1)555 1528 y Fq(A)612 1513 y Fx(\))27 b(=)h(1)830 1528 y Fr(F)7 b Fn(\()p Fq(A)p Fn(\))1031 1513 y Fx(for)32 b(all)f Ft(A)c Fu(2)h Fx(Ob)17 b Fu(C)6 b Fx(,)149 1629 y(2.)42 b Fu(F)339 1644 y Fq(A;C)470 1629 y Fx(\()p Ft(g)t(f)11 b Fx(\))27 b(=)g Fu(F)858 1644 y Fq(B)s(;C)993 1629 y Fx(\()p Ft(g)t Fx(\))p Fu(F)1192 1644 y Fq(A;B)1324 1629 y Fx(\()p Ft(f)11 b Fx(\))32 b(for)g(all)e Ft(A;)17 b(B)5 b(;)17 b(C)35 b Fu(2)28 b Fx(Ob)17 b Fu(C)6 b Ft(:)267 1745 y Fx([)p Fu(F)366 1760 y Fq(A;C)497 1745 y Fx(\()p Ft(g)t(f)11 b Fx(\))27 b(=)g Fu(F)885 1760 y Fq(A;B)1018 1745 y Fx(\()p Ft(f)11 b Fx(\))p Fu(F)1225 1760 y Fq(B)s(;C)1359 1745 y Fx(\()p Ft(g)t Fx(\))32 b(for)g(all)f Ft(A;)17 b(B)5 b(;)17 b(C)34 b Fu(2)28 b Fx(Ob)17 b Fu(C)6 b Fx(].)0 1889 y Fv(Notation:)42 b Fx(W)-8 b(e)33 b(write)1046 2006 y Ft(A)28 b Fu(2)g(C)89 b Fx(instead)33 b(of)219 b Ft(A)28 b Fu(2)g Fx(Ob)17 b Fu(C)1053 2123 y Ft(f)39 b Fu(2)28 b(C)96 b Fx(instead)33 b(of)82 b Ft(f)39 b Fu(2)28 b Fx(Mor)2237 2138 y Fr(C)2282 2123 y Fx(\()p Ft(A;)17 b(B)5 b Fx(\))1065 2239 y Fu(F)10 b Fx(\()p Ft(f)h Fx(\))100 b(instead)33 b(of)236 b Fu(F)2106 2254 y Fq(A;B)2239 2239 y Fx(\()p Ft(f)11 b Fx(\).)149 2418 y Fv(Examples)37 b(8.2.2.)199 b Fx(1.)41 b(Id)28 b(:)g Fv(Set)g Fu(\000)-60 b(!)27 b Fv(Set)149 2534 y Fx(2.)42 b(F)-8 b(orget)32 b(:)43 b Ft(R)q Fx(-)p Fv(Mo)s(d)28 b Fu(\000)-60 b(!)28 b Fv(Set)149 2650 y Fx(3.)42 b(F)-8 b(orget)32 b(:)43 b Fv(Ri)27 b Fu(\000)-60 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3414 y Fv(Lemma)38 b(8.2.3.)198 b Fx(1.)42 b Fs(L)-5 b(et)35 b Ft(X)g Fu(2)28 b(C)6 b Fs(.)45 b(Then)1061 3586 y Fx(Ob)17 b Fu(C)34 b(3)28 b Ft(A)g Fu(7!)f Fx(Mor)1792 3601 y Fr(C)1837 3586 y Fx(\()p Ft(X)r(;)17 b(A)p Fx(\))28 b Fu(2)g Fx(Ob)17 b Fv(Set)346 3772 y Fx(Mor)522 3787 y Fr(C)567 3772 y Fx(\()p Ft(A;)g(B)5 b Fx(\))28 b Fu(3)g Ft(f)38 b Fu(7!)27 b Ft(M)10 b(or)1369 3787 y Fr(C)1415 3772 y Fx(\()p Ft(X)r(;)17 b(f)11 b Fx(\))27 b Fu(2)i Fx(Mor)1975 3787 y Fj(Set)2093 3772 y Fx(\(Mor)2307 3787 y Fr(C)2352 3772 y Fx(\()p Ft(X)r(;)17 b(A)p Fx(\))p Ft(;)g Fx(Mor)2848 3787 y Fq(C)2907 3772 y Fx(\()p Ft(X)r(;)g(B)5 b Fx(\)\))p Ft(;)267 3923 y Fs(with)38 b Fx(Mor)658 3938 y Fq(C)717 3923 y Fx(\()p Ft(X)r(;)17 b(f)11 b Fx(\))34 b(:)g(Mor)1250 3938 y Fq(C)1309 3923 y Fx(\()p Ft(X)r(;)17 b(A)p Fx(\))35 b Fu(3)f Ft(g)k Fu(7!)33 b Ft(f)11 b(g)37 b Fu(2)e Fx(Mor)2358 3938 y Fq(C)2417 3923 y Fx(\()p Ft(X)r(;)17 b(B)5 b Fx(\))39 b Fs(or)f Fx(Mor)3043 3938 y Fq(C)3102 3923 y Fx(\()p Ft(X)r(;)17 b(f)11 b Fx(\)\()p Ft(g)t Fx(\))33 b(=)267 4039 y Ft(f)11 b(g)38 b Fs(is)c(a)h(c)-5 b(ovariant)34 b(functor)h Fx(Mor)1543 4054 y Fq(C)1602 4039 y Fx(\()p Ft(X)r(;)17 b Fs(-)p Fx(\))p Fs(.)149 4156 y Fx(2.)42 b Fs(L)-5 b(et)35 b Ft(X)h Fu(2)28 b Ft(C)7 b Fs(.)44 b(Then)1059 4327 y Fx(Ob)16 b Fu(C)34 b(3)28 b Ft(A)g Fu(7!)f Fx(Mor)1789 4342 y Fr(C)1835 4327 y Fx(\()p Ft(A;)17 b(X)8 b Fx(\))27 b Fu(2)h Fx(Ob)16 b Fv(Set)371 4514 y Fx(Mor)547 4529 y Fr(C)592 4514 y Fx(\()p Ft(A;)h(B)5 b Fx(\))28 b Fu(3)g Ft(f)38 b Fu(7!)27 b Fx(Mor)1375 4529 y Fr(C)1420 4514 y Fx(\()p Ft(f)5 b(;)17 b(X)8 b Fx(\))28 b Fu(2)g Fx(Mor)1980 4529 y Fj(Set)2098 4514 y Fx(\(Mor)2312 4529 y Fr(C)2357 4514 y Fx(\()p Ft(B)5 b(;)17 b(X)8 b Fx(\))p Ft(;)17 b Fx(Mor)2865 4529 y Fr(C)2910 4514 y Fx(\()p Ft(A;)g(X)8 b Fx(\)\))267 4665 y Fs(with)27 b Fx(Mor)647 4680 y Fr(C)692 4665 y Fx(\()p Ft(f)5 b(;)17 b(X)8 b Fx(\))27 b(:)h(Mor)1212 4680 y Fr(C)1257 4665 y Fx(\()p Ft(B)5 b(;)17 b(X)8 b Fx(\))28 b Fu(3)g Ft(g)j Fu(7!)c Ft(g)t(f)38 b Fu(2)28 b Fx(Mor)2279 4680 y Fr(C)2324 4665 y Fx(\()p Ft(A;)17 b(X)8 b Fx(\))27 b Fs(or)g Fx(Mor)2927 4680 y Fr(C)2972 4665 y Fx(\()p Ft(f)5 b(;)17 b(X)8 b Fx(\)\()p Ft(g)t Fx(\))26 b(=)i Ft(g)t(f)267 4781 y Fs(is)34 b(a)h(c)-5 b(ontr)g(avariant)34 b(functor)h Fx(Mor)1572 4796 y Fr(C)1618 4781 y Fx(\()p Fs(-)o Ft(;)17 b(X)8 b Fx(\))p Fs(.)149 4964 y Fi(Pr)n(oof.)42 b Fx(1.)79 b(Mor)865 4979 y Fr(C)910 4964 y Fx(\()p Ft(X)r(;)17 b Fx(1)1124 4979 y Fq(A)1181 4964 y Fx(\)\()p Ft(g)t Fx(\))46 b(=)i(1)1565 4979 y Fq(A)1621 4964 y Ft(g)j Fx(=)d Ft(g)i Fx(=)e(id)o(\()p Ft(g)t Fx(\))p Ft(;)60 b Fx(Mor)2535 4979 y Fq(C)2594 4964 y Fx(\()p Ft(X)r(;)17 b(f)11 b Fx(\)Mor)3032 4979 y Fq(C)3091 4964 y Fx(\()p Ft(X)r(;)17 b(g)t Fx(\)\()p Ft(h)p Fx(\))47 b(=)0 5080 y Ft(f)11 b(g)t(h)27 b Fx(=)g(Mor)472 5095 y Fq(C)532 5080 y Fx(\()p Ft(X)r(;)17 b(f)11 b(g)t Fx(\)\()p Ft(h)p Fx(\).)149 5196 y(2.)44 b(analogously)-8 b(.)p 3529 5196 4 66 v 3533 5134 59 4 v 3533 5196 V 3591 5196 4 66 v eop %%Page: 6 6 6 5 bop 0 236 a Fo(6)1494 b(8.)77 b(TOOLBO)n(X)149 425 y Fv(Remark)37 b(8.2.4.)50 b Fx(The)33 b(preceding)h(lemma)c(sho)m(ws)k (that)f(Mor)2507 440 y Fr(C)2552 425 y Fx(\(-)p Ft(;)17 b Fx(-)o(\))32 b(is)h(a)f(functor)h(in)f(b)s(oth)0 541 y(argumen)m(ts.)68 b(A)41 b(functor)g(in)f(t)m(w)m(o)i(argumen)m(ts)f (is)f(called)g(a)g Fs(bifunctor)p Fx(.)68 b(W)-8 b(e)41 b(can)g(regards)h(the)0 658 y(bifunctor)32 b(Mor)596 673 y Fr(C)641 658 y Fx(\(-)p Ft(;)17 b Fx(-)o(\))32 b(as)h(a)f(co)m(v)-5 b(arian)m(t)32 b(functor)1222 815 y(Mor)1398 830 y Fr(C)1443 815 y Fx(\(-)o Ft(;)17 b Fx(-)o(\))28 b(:)g Fu(C)1768 773 y Fq(op)1864 815 y Fu(\002)23 b(C)34 b(\000)-60 b(!)27 b Fv(Set)p Ft(:)0 971 y Fx(The)40 b(use)h(of)e(the)g (dual)g(category)h(remo)m(v)m(es)g(the)g(fact)f(that)h(the)f(bifunctor) g(Mor)3047 986 y Fr(C)3092 971 y Fx(\(-)p Ft(;)17 b Fx(-)o(\))39 b(is)g(con-)0 1088 y(tra)m(v)-5 b(arian)m(t)32 b(in)f(the)i(\014rst)g (v)-5 b(ariable.)149 1204 y(Ob)m(viously)28 b(the)f(comp)s(osition)e (of)i(t)m(w)m(o)h(functors)f(is)g(again)f(a)h(functor)g(and)g(this)g (comp)s(osition)0 1320 y(is)32 b(asso)s(ciativ)m(e.)43 b(F)-8 b(urthermore)32 b(for)g(eac)m(h)i(category)f Fu(C)38 b Fx(there)c(is)e(an)g(iden)m(tit)m(y)h(functor)f(Id)3311 1335 y Fr(C)3356 1320 y Fx(.)149 1436 y(F)-8 b(unctors)27 b(of)f(the)h(form)e(Mor)1213 1451 y Fr(C)1258 1436 y Fx(\()p Ft(X)r(;)17 b Fx(-\))26 b(resp.)43 b(Mor)1939 1451 y Fr(C)1984 1436 y Fx(\(-)o Ft(;)17 b(X)8 b Fx(\))26 b(are)h(called)e Fs(r)-5 b(epr)g(esentable)28 b(functors)0 1553 y Fx(\(co)m(v)-5 b(arian)m(t)23 b(resp.)42 b(con)m(tra)m(v)-5 b(arian)m(t\))23 b(and)i Ft(X)31 b Fx(is)24 b(called)f(the)h Fs(r)-5 b(epr)g(esenting)26 b(obje)-5 b(ct)24 b Fx(\(see)h(also)e (section)0 1669 y(8.8\).)p eop %%Page: 7 7 7 6 bop 1137 236 a Fo(3.)77 b(NA)-6 b(TURAL)31 b(TRANSF)n(ORMA)-6 b(TIONS)1070 b(7)1103 425 y Fv(3.)94 b(Natural)36 b(T)-9 b(ransformations)149 600 y(De\014nition)37 b(8.3.1.)49 b Fx(Let)j Fu(F)72 b Fx(:)61 b Fu(C)68 b(\000)-60 b(!)62 b(D)55 b Fx(and)d Fu(G)68 b Fx(:)62 b Fu(C)68 b(\000)-60 b(!)61 b(D)55 b Fx(b)s(e)e(t)m(w)m(o)g(functors.)104 b(A)0 716 y Fs(natur)-5 b(al)31 b(tr)-5 b(ansformation)28 b Fx(or)g(a)h Fs(functorial)h(morphism)e Ft(')f Fx(:)h Fu(F)37 b(\000)-60 b(!)28 b(G)35 b Fx(is)28 b(a)g(family)e(of)i (morphisms)0 832 y Fu(f)p Ft(')p Fx(\()p Ft(A)p Fx(\))f(:)h Fu(F)10 b Fx(\()p Ft(A)p Fx(\))27 b Fu(\000)-60 b(!)28 b(G)6 b Fx(\()p Ft(A)p Fx(\))p Fu(j)p Ft(A)28 b Fu(2)g(C)6 b(g)32 b Fx(suc)m(h)i(that)f(the)g(diagram)1349 1606 y Fu(F)10 b Fx(\()p Ft(B)5 b Fx(\))453 b Fu(G)6 b Fx(\()p Ft(B)f Fx(\))p 1614 1583 397 4 v 1928 1581 a Fl(-)1703 1676 y Ft(')p Fx(\()p Ft(B)g Fx(\))1352 1119 y Fu(F)10 b Fx(\()p Ft(A)p Fx(\))459 b Fu(G)6 b Fx(\()p Ft(A)p Fx(\))p 1611 1096 403 4 v 1931 1094 a Fl(-)1706 1050 y Ft(')p Fx(\()p Ft(A)p Fx(\))p 1465 1513 4 351 v 1467 1513 a Fl(?)1212 1363 y Fu(F)k Fx(\()p Ft(f)h Fx(\))p 2148 1513 V 2150 1513 a Fl(?)2189 1363 y Fu(G)6 b Fx(\()p Ft(f)11 b Fx(\))0 1893 y(comm)m(utes)32 b(for)h(all)d Ft(f)38 b Fx(:)28 b Ft(A)g Fu(\000)-60 b(!)28 b Ft(B)37 b Fx(in)32 b Fu(C)6 b Fx(,)33 b(i.e.)43 b Fu(G)6 b Fx(\()p Ft(f)11 b Fx(\))p Ft(')p Fx(\()p Ft(A)p Fx(\))27 b(=)h Ft(')p Fx(\()p Ft(B)5 b Fx(\))p Fu(F)10 b Fx(\()p Ft(f)h Fx(\).)149 2107 y Fv(Lemma)38 b(8.3.2.)49 b Fs(Given)g(c)-5 b(ovariant)49 b(functors)h Fu(F)65 b Fx(=)55 b(Id)2350 2122 y Fj(Set)2524 2107 y Fx(:)g Fv(Set)g Fu(\000)-58 b(!)56 b Fv(Set)50 b Fs(and)f Fu(G)61 b Fx(=)0 2223 y(Mor)176 2238 y Fj(Set)294 2223 y Fx(\(Mor)508 2238 y Fj(Set)627 2223 y Fx(\()p Fu(\000)p Ft(;)17 b(A)p Fx(\))p Ft(;)g(A)p Fx(\))27 b(:)h Fv(Set)g Fu(\000)-58 b(!)28 b Fv(Set)35 b Fs(for)f(a)h(set)g Ft(A)p Fs(.)45 b(Then)34 b Ft(')28 b Fx(:)f Fu(F)38 b(\000)-58 b(!)28 b(G)41 b Fs(with)605 2449 y Ft(')p Fx(\()p Ft(B)5 b Fx(\))28 b(:)g Ft(B)33 b Fu(3)28 b Ft(b)g Fu(7!)f Fx(\(Mor)1518 2464 y Fj(Set)1636 2449 y Fx(\()p Ft(B)5 b(;)17 b(A)p Fx(\))28 b Fu(3)g Ft(f)38 b Fu(7!)28 b Ft(f)11 b Fx(\()p Ft(b)p Fx(\))27 b Fu(2)i Ft(A)p Fx(\))e Fu(2)h(G)6 b Fx(\()p Ft(B)f Fx(\))0 2675 y Fs(is)35 b(a)f(natur)-5 b(al)35 b(tr)-5 b(ansformation.)149 2889 y Fi(Pr)n(oof.)42 b Fx(Giv)m(en)33 b Ft(g)e Fx(:)d Ft(B)k Fu(\000)-60 b(!)28 b Ft(C)7 b Fx(.)44 b(Then)33 b(the)g(follo)m(wing)d(diagram)h(comm)m(utes)786 3170 y Ft(B)414 b Fx(Mor)1450 3185 y Fj(Set)1568 3170 y Fx(\(Mor)1782 3185 y Fj(Set)1901 3170 y Fx(\()p Ft(B)5 b(;)17 b(A)p Fx(\))p Ft(;)g(A)p Fx(\))p 894 3147 351 4 v 1162 3145 a Fl(-)960 3101 y Ft(')p Fx(\()p Ft(B)5 b Fx(\))787 3658 y Ft(C)421 b Fx(Mor)1454 3673 y Fj(Set)1572 3658 y Fx(\(Mor)1786 3673 y Fj(Set)1904 3658 y Fx(\()p Ft(C)r(;)17 b(A)p Fx(\))p Ft(;)g(A)p Fx(\))p 893 3635 356 4 v 1166 3633 a Fl(-)962 3589 y Ft(')p Fx(\()p Ft(C)7 b Fx(\))p 824 3565 4 351 v 826 3565 a Fl(?)736 3401 y Ft(g)p 1799 3565 V 1800 3565 a Fl(?)1839 3414 y Fx(Mor)2016 3429 y Fj(Set)2134 3414 y Fx(\(Mor)2348 3429 y Fj(Set)2466 3414 y Fx(\()p Ft(g)t(;)17 b(A)p Fx(\))p Ft(;)g(A)p Fx(\))0 3867 y(since)632 4066 y Ft(')p Fx(\()p Ft(C)7 b Fx(\))p Fu(F)j Fx(\()p Ft(g)t Fx(\)\()p Ft(b)p Fx(\)\()p Ft(f)h Fx(\))26 b(=)h Ft(')p Fx(\()p Ft(C)7 b Fx(\))p Ft(g)t Fx(\()p Ft(b)p Fx(\)\()p Ft(f)k Fx(\))27 b(=)g Ft(f)11 b(g)t Fx(\()p Ft(b)p Fx(\))28 b(=)f Ft(')p Fx(\()p Ft(B)5 b Fx(\)\()p Ft(b)p Fx(\)\()p Ft(f)11 b(g)t Fx(\))366 4182 y(=)28 b([)p Ft(')p Fx(\()p Ft(B)5 b Fx(\)\()p Ft(b)p Fx(\)Mor)1009 4197 y Fj(Set)1127 4182 y Fx(\()p Ft(g)t(;)17 b(A)p Fx(\)]\()p Ft(f)11 b Fx(\))27 b(=)g([Mor)1867 4197 y Fj(Set)1985 4182 y Fx(\(Mor)2199 4197 y Fj(Set)2317 4182 y Fx(\()p Ft(g)t(;)17 b(A)p Fx(\))p Ft(;)g(A)p Fx(\))p Ft(')p Fx(\()p Ft(A)p Fx(\)\()p Ft(b)p Fx(\)]\()p Ft(f)11 b Fx(\))p Ft(:)p 3529 4401 4 66 v 3533 4339 59 4 v 3533 4401 V 3591 4401 4 66 v 149 4615 a Fv(Lemma)38 b(8.3.3.)49 b Fs(L)-5 b(et)25 b Ft(f)39 b Fx(:)27 b Ft(A)h Fu(\000)-58 b(!)28 b Ft(B)i Fs(b)-5 b(e)25 b(a)g(morphism)e(in)i Fu(C)6 b Fs(.)41 b(Then)25 b Fx(Mor)2797 4630 y Fr(C)2842 4615 y Fx(\()p Ft(f)5 b(;)17 b Fs(-)p Fx(\))27 b(:)h(Mor)3308 4630 y Fr(C)3353 4615 y Fx(\()p Ft(B)5 b(;)17 b Fs(-)p Fx(\))27 b Fu(\000)-58 b(!)0 4731 y Fx(Mor)176 4746 y Fr(C)221 4731 y Fx(\()p Ft(A;)17 b Fs(-)p Fx(\))40 b Fs(given)f(by)i Fx(Mor)1057 4746 y Fr(C)1102 4731 y Fx(\()p Ft(f)5 b(;)17 b(C)7 b Fx(\))37 b(:)h(Mor)1630 4746 y Fr(C)1675 4731 y Fx(\()p Ft(B)5 b(;)17 b(C)7 b Fx(\))38 b Fu(3)g Ft(g)j Fu(7!)c Ft(g)t(f)48 b Fu(2)38 b Fx(Mor)2745 4746 y Fr(C)2790 4731 y Fx(\()p Ft(A;)17 b(C)7 b Fx(\))40 b Fs(is)g(a)g(natur)-5 b(al)0 4848 y(tr)g(ansformation)34 b(of)h(c)-5 b(ovariant)34 b(functors.)149 4964 y(L)-5 b(et)36 b Ft(f)k Fx(:)29 b Ft(A)h Fu(\000)-58 b(!)29 b Ft(B)41 b Fs(b)-5 b(e)35 b(a)h(morphism)e(in)h Fu(C)6 b Fs(.)47 b(Then)35 b Fx(Mor)2175 4979 y Fr(C)2220 4964 y Fx(\()p Fs(-)p Ft(;)17 b(f)11 b Fx(\))29 b(:)g(Mor)2695 4979 y Fr(C)2740 4964 y Fx(\()p Fs(-)p Ft(;)17 b(A)p Fx(\))29 b Fu(\000)-58 b(!)29 b Fx(Mor)3321 4979 y Fr(C)3366 4964 y Fx(\()p Fs(-)p Ft(;)17 b(B)5 b Fx(\))0 5080 y Fs(given)24 b(by)i Fx(Mor)538 5095 y Fr(C)583 5080 y Fx(\()p Ft(C)r(;)17 b(f)11 b Fx(\))26 b(:)i(Mor)1091 5095 y Fr(C)1136 5080 y Fx(\()p Ft(C)r(;)17 b(A)p Fx(\))27 b Fu(3)h Ft(g)j Fu(7!)c Ft(f)11 b(g)31 b Fu(2)d Fx(Mor)2134 5095 y Fr(C)2179 5080 y Fx(\()p Ft(C)r(;)17 b(B)5 b Fx(\))25 b Fs(is)g(a)g(natur)-5 b(al)25 b(tr)-5 b(ansformation)0 5196 y(of)35 b(c)-5 b(ontr)g(avariant)34 b(functors.)p eop %%Page: 8 8 8 7 bop 0 236 a Fo(8)1494 b(8.)77 b(TOOLBO)n(X)149 425 y Fi(Pr)n(oof.)42 b Fx(Let)33 b Ft(h)28 b Fx(:)g Ft(C)34 b Fu(\000)-60 b(!)28 b Ft(C)1174 389 y Fr(0)1230 425 y Fx(b)s(e)33 b(a)f(morphism)f(in)h Fu(C)6 b Fx(.)43 b(Then)34 b(the)f(diagrams)1054 1133 y(Mor)1230 1148 y Fr(C)1275 1133 y Fx(\()p Ft(B)5 b(;)17 b(C)1513 1096 y Fr(0)1536 1133 y Fx(\))458 b(Mor)2208 1148 y Fr(C)2253 1133 y Fx(\()p Ft(A;)17 b(C)2485 1096 y Fr(0)2508 1133 y Fx(\))p 1603 1110 401 4 v 1921 1108 a Fl(-)1624 1178 y Fn(Mor)1751 1189 y Fh(C)1792 1178 y Fn(\()p Fq(f)s(;C)1931 1154 y Fh(0)1953 1178 y Fn(\))1066 645 y Fx(Mor)1242 660 y Fr(C)1287 645 y Fx(\()p Ft(B)5 b(;)17 b(C)7 b Fx(\))481 b(Mor)2220 660 y Fr(C)2265 645 y Fx(\()p Ft(A;)17 b(C)7 b Fx(\))p 1591 622 425 4 v 1933 620 a Fl(-)1635 584 y Fn(Mor)1763 595 y Fh(C)1803 584 y Fn(\()p Fq(f)s(;C)e Fn(\))p 1312 1039 4 351 v 1314 1039 a Fl(?)936 881 y Fn(Mor)1063 892 y Fh(C)1103 881 y Fn(\()p Fq(B)s(;h)p Fn(\))p 2287 1039 V 2289 1039 a Fl(?)2328 881 y Fn(Mor)2455 892 y Fh(C)2496 881 y Fn(\()p Fq(A;h)p Fn(\))0 1352 y Fx(and)1068 2000 y(Mor)1244 2015 y Fr(C)1290 2000 y Fx(\()p Ft(C)r(;)17 b(A)p Fx(\))485 b(Mor)2216 2015 y Fr(C)2261 2000 y Fx(\()p Ft(C)r(;)17 b(B)5 b Fx(\))p 1582 1977 429 4 v 1928 1975 a Fl(-)1629 2044 y Fn(Mor)1757 2055 y Fh(C)1797 2044 y Fn(\()p Fq(C)q(;f)i Fn(\))1054 1512 y Fx(Mor)1230 1527 y Fr(C)1275 1512 y Fx(\()p Ft(C)1390 1476 y Fr(0)1413 1512 y Ft(;)17 b(A)p Fx(\))458 b(Mor)2202 1527 y Fr(C)2247 1512 y Fx(\()p Ft(C)2362 1476 y Fr(0)2385 1512 y Ft(;)17 b(B)5 b Fx(\))p 1597 1489 401 4 v 1915 1487 a Fl(-)1616 1452 y Fn(Mor)1744 1463 y Fh(C)1784 1452 y Fn(\()p Fq(C)1866 1429 y Fh(0)1889 1452 y Fq(;f)i Fn(\))p 1309 1907 4 351 v 1311 1907 a Fl(?)936 1748 y Fn(Mor)1064 1759 y Fh(C)1104 1748 y Fn(\()p Fq(h;A)p Fn(\))p 2284 1907 V 2286 1907 a Fl(?)2325 1748 y Fn(Mor)2452 1759 y Fh(C)2493 1748 y Fn(\()p Fq(h;B)s Fn(\))0 2196 y Fx(comm)m(ute.)p 3529 2196 4 66 v 3533 2133 59 4 v 3533 2196 V 3591 2196 4 66 v 149 2394 a Fv(Remark)37 b(8.3.4.)50 b Fx(The)35 b(comp)s(osition)d(of)i(t)m(w)m(o)h(natural)f (transformations)e(is)i(again)f(a)i(nat-)0 2510 y(ural)d (transformation.)41 b(The)33 b(iden)m(tit)m(y)g(id)1545 2525 y Fr(F)1606 2510 y Fx(\()p Ft(A)p Fx(\))28 b(:=)g(1)1963 2525 y Fr(F)7 b Fn(\()p Fq(A)p Fn(\))2164 2510 y Fx(is)32 b(also)g(a)g(natural)f(transformation.)149 2708 y Fv(De\014nition)37 b(8.3.5.)49 b Fx(A)37 b(natural)e(transformation)g Ft(')g Fx(:)g Fu(F)44 b(\000)-60 b(!)35 b(G)43 b Fx(is)37 b(called)e(a)i Fs(natur)-5 b(al)39 b(iso-)0 2824 y(morphism)33 b Fx(if)g(there)h (exists)h(a)f(natural)f(transformation)f Ft( )i Fx(:)d Fu(G)36 b(\000)-60 b(!)30 b(F)44 b Fx(suc)m(h)35 b(that)f Ft(')23 b Fu(\016)g Ft( )35 b Fx(=)30 b(id)3550 2839 y Fr(G)0 2940 y Fx(and)39 b Ft( )31 b Fu(\016)26 b Ft(')38 b Fx(=)h(id)664 2955 y Fr(F)725 2940 y Fx(.)63 b(The)40 b(natural)e(transformation)f Ft( )43 b Fx(is)38 b(uniquely)h (determined)g(b)m(y)h Ft(')p Fx(.)63 b(W)-8 b(e)0 3056 y(write)32 b Ft(')313 3020 y Fr(\000)p Fn(1)435 3056 y Fx(:=)c Ft( )t Fx(.)149 3172 y(A)46 b(functor)g Fu(F)55 b Fx(is)45 b(said)g(to)g(b)s(e)h Fs(isomorphic)e Fx(to)i(a)f(functor)h Fu(G)52 b Fx(if)44 b(there)i(exists)h(a)e(natural)0 3289 y(isomorphism)30 b Ft(')e Fx(:)f Fu(F)38 b(\000)-60 b(!)27 b(G)6 b Fx(.)149 3486 y Fv(Problem)36 b(8.3.2.)49 b Fx(1.)57 b(Let)37 b Fu(F)10 b Ft(;)17 b Fu(G)41 b Fx(:)35 b Fu(C)41 b(\000)-60 b(!)36 b(D)j Fx(b)s(e)f(functors.)57 b(Sho)m(w)38 b(that)f(a)f(natural)g(trans-)0 3603 y(formation)27 b Ft(')h Fx(:)g Fu(F)37 b(\000)-60 b(!)28 b(G)36 b Fx(is)29 b(a)h(natural)e(isomorphism)g(if)g(and)i(only)g(if)f Ft(')p Fx(\()p Ft(A)p Fx(\))g(is)h(an)f(isomorphism)0 3719 y(for)j(all)f(ob)5 b(jects)33 b Ft(A)28 b Fu(2)g(C)6 b Fx(.)149 3835 y(2.)51 b(Let)35 b(\()p Ft(A)24 b Fu(\002)g Ft(B)5 b(;)17 b(p)861 3850 y Fq(A)918 3835 y Ft(;)g(p)1011 3850 y Fq(B)1071 3835 y Fx(\))35 b(b)s(e)g(the)h(pro)s(duct)f(of)g Ft(A)g Fx(and)g Ft(B)40 b Fx(in)34 b Fu(C)6 b Fx(.)51 b(Then)36 b(there)g(is)e(a)h(natural)0 3951 y(isomorphism)909 4108 y(Mor\(-)o Ft(;)17 b(A)22 b Fu(\002)h Ft(B)5 b Fx(\))1539 4081 y Fu(\030)1539 4112 y Fx(=)1644 4108 y(Mor)1820 4123 y Fr(C)1865 4108 y Fx(\(-)o Ft(;)17 b(A)p Fx(\))22 b Fu(\002)h Fx(Mor)2388 4123 y Fr(C)2433 4108 y Fx(\(-)o Ft(;)17 b(B)5 b Fx(\))p Ft(:)149 4286 y Fx(3.)62 b(Let)39 b Fu(C)45 b Fx(b)s(e)39 b(a)g(category)g(with)g(\014nite)f(pro)s (ducts.)63 b(F)-8 b(or)38 b(eac)m(h)i(ob)5 b(ject)39 b Ft(A)g Fx(in)f Fu(C)45 b Fx(sho)m(w)40 b(that)0 4402 y(there)34 b(exists)g(a)f(morphism)e(\001)1139 4417 y Fq(A)1225 4402 y Fx(:)e Ft(A)g Fu(\000)-60 b(!)28 b Ft(A)23 b Fu(\002)g Ft(A)33 b Fx(satisfying)f Ft(p)2314 4417 y Fn(1)2354 4402 y Fx(\001)2435 4417 y Fq(A)2521 4402 y Fx(=)c(1)2674 4417 y Fq(A)2760 4402 y Fx(=)g Ft(p)2913 4417 y Fn(2)2953 4402 y Fx(\001)3034 4417 y Fq(A)3091 4402 y Fx(.)45 b(Sho)m(w)34 b(that)0 4518 y(this)e(de\014nes)j(a)d (natural)f(transformation.)42 b(What)32 b(are)h(the)g(functors?)149 4634 y(4.)87 b(Let)47 b Fu(C)54 b Fx(b)s(e)47 b(a)g(category)g(with)g (\014nite)g(pro)s(ducts.)88 b(Sho)m(w)48 b(that)e(there)i(is)f(a)g(bi)p Fs(functor)0 4751 y Fx(-)22 b Fu(\002)h Fx(-)29 b(:)g Fu(C)g(\002)23 b(C)35 b(\000)-60 b(!)29 b(C)39 b Fx(suc)m(h)c(that)e (\(-)22 b Fu(\002)i Fx(-)o(\)\()p Ft(A;)17 b(B)5 b Fx(\))33 b(is)g(the)h(ob)5 b(ject)34 b(of)f(a)g(pro)s(duct)g(of)g Ft(A)g Fx(and)h Ft(B)5 b Fx(.)46 b(W)-8 b(e)0 4867 y(denote)33 b(elemen)m(ts)f(in)g(the)g(image)f(of)g(this)h(functor)h(b)m(y)g Ft(A)21 b Fu(\002)h Ft(B)33 b Fx(:=)27 b(\(-)21 b Fu(\002)h Fx(-)o(\)\()p Ft(A;)17 b(B)5 b Fx(\))32 b(and)h(similarly)0 4983 y Ft(f)g Fu(\002)22 b Ft(g)t Fx(.)149 5099 y(5.)90 b(With)48 b(the)g(notation)f(of)h(the)g(preceding)h(problem)e(sho)m(w)i (that)f(there)h(is)e(a)h(natural)0 5216 y(transformation)36 b Ft(\013)q Fx(\()p Ft(A;)17 b(B)5 b(;)17 b(C)7 b Fx(\))37 b(:)h(\()p Ft(A)26 b Fu(\002)h Ft(B)5 b Fx(\))26 b Fu(\002)h Ft(C)1831 5188 y Fu(\030)1831 5220 y Fx(=)1946 5216 y Ft(A)f Fu(\002)h Fx(\()p Ft(B)k Fu(\002)c Ft(C)7 b Fx(\).)61 b(Sho)m(w)39 b(that)f(the)h(diagram)p eop %%Page: 9 9 9 8 bop 1137 236 a Fo(3.)77 b(NA)-6 b(TURAL)31 b(TRANSF)n(ORMA)-6 b(TIONS)1070 b(9)0 425 y Fx(\()p Fs(c)-5 b(oher)g(enc)g(e)34 b(or)g(c)-5 b(onstr)g(aints)p Fx(\))-1 610 y(\(\()p Ft(A)23 b Fu(\002)f Ft(B)5 b Fx(\))22 b Fu(\002)h Ft(C)7 b Fx(\))22 b Fu(\002)h Ft(D)635 b Fx(\()p Ft(A)22 b Fu(\002)h Fx(\()p Ft(B)k Fu(\002)22 b Ft(C)7 b Fx(\)\))22 b Fu(\002)h Ft(D)p 858 587 576 4 v 1351 585 a Fl(-)949 549 y Fq(\013)p Fn(\()p Fq(A;B)s(;C)5 b Fn(\))p Fr(\002)p Fn(1)2924 610 y Ft(A)22 b Fu(\002)h Fx(\(\()p Ft(B)k Fu(\002)c Ft(C)7 b Fx(\))22 b Fu(\002)h Ft(D)s Fx(\))p 2320 587 V 2813 585 a Fl(-)2401 549 y Fq(\013)p Fn(\()p Fq(A;B)s Fr(\002)p Fq(C)q(;D)r Fn(\))p 413 907 4 254 v 414 907 a Fl(?)453 796 y Fq(\013)p Fn(\()p Fq(A)p Fr(\002)p Fq(B)s(;C)q(;D)r Fn(\))p 3338 907 V 3339 907 a Fl(?)2903 796 y Fn(1)p Fr(\002)p Fq(\013)p Fn(\()p Fq(B)s(;C)q(;D)r Fn(\))-1 1000 y Fx(\()p Ft(A)g Fu(\002)f Ft(B)5 b Fx(\))22 b Fu(\002)h Fx(\()p Ft(C)29 b Fu(\002)23 b Ft(D)s Fx(\))2094 b Ft(A)22 b Fu(\002)h Fx(\()p Ft(B)k Fu(\002)c Fx(\()p Ft(C)29 b Fu(\002)23 b Ft(D)s Fx(\)\))p 858 977 2038 4 v 2813 975 a Fl(-)1668 939 y Fq(\013)p Fn(\()p Fq(A;B)s(;C)5 b Fr(\002)p Fq(D)r Fn(\))0 1135 y Fx(comm)m(utes.)149 1251 y(6.)77 b(With)44 b(the)g(notation)e(of)i(the)g(preceding)g(problem)f(sho)m(w)i(that)e (there)i(are)f(a)f(natural)0 1367 y(transformations)35 b Ft(\025)p Fx(\()p Ft(A)p Fx(\))f(:)g Ft(E)d Fu(\002)25 b Ft(A)34 b Fu(\000)-60 b(!)35 b Ft(A)h Fx(and)h Ft(\032)p Fx(\()p Ft(A)p Fx(\))d(:)g Ft(A)25 b Fu(\002)g Ft(E)41 b Fu(\000)-60 b(!)34 b Ft(A)i Fx(suc)m(h)i(that)e(the)h(diagram)0 1484 y(\()p Fs(c)-5 b(oher)g(enc)g(e)34 b(or)g(c)-5 b(onstr)g(aints)p Fx(\))940 1668 y(\()p Ft(A)22 b Fu(\002)h Ft(E)6 b Fx(\))22 b Fu(\002)h Ft(B)625 b(A)22 b Fu(\002)h Fx(\()p Ft(E)28 b Fu(\002)23 b Ft(B)5 b Fx(\))p 1518 1645 564 4 v 1999 1643 a Fl(-)1648 1607 y Fq(\013)p Fn(\()p Fq(A;E)t(;B)s Fn(\))1663 2063 y Ft(A)22 b Fu(\002)h Ft(B)1212 1855 y Fq(\032)p Fn(\()p Fq(A)p Fn(\))p Fr(\002)p Fn(1)1317 1767 y Fl(Q)1400 1822 y(Q)1483 1878 y(Q)1566 1933 y(Q)1615 1965 y(Q)-83 b(s)2154 1855 y Fn(1)p Fr(\002)p Fq(\025)p Fn(\()p Fq(B)s Fn(\))2200 1767 y Fl(\021)2117 1822 y(\021)2034 1878 y(\021)1951 1933 y(\021)1902 1965 y(\021)g(+)149 2208 y Fv(De\014nition)37 b(8.3.6.)49 b Fx(Let)32 b Fu(C)38 b Fx(and)31 b Fu(D)k Fx(b)s(e)d(categories.)43 b(A)32 b(co)m(v)-5 b(arian)m(t)31 b(functor)h Fu(F)37 b Fx(:)27 b Fu(C)34 b(\000)-60 b(!)28 b(D)34 b Fx(is)0 2324 y(called)f(an)h Fs(e)-5 b(quivalenc)g(e)35 b(of)h(c)-5 b(ate)g(gories)33 b Fx(if)g(there)i(exists)f(a)g(co)m(v)-5 b(arian)m(t)34 b(functor)g Fu(G)i Fx(:)31 b Fu(D)h(\000)-60 b(!)31 b(C)40 b Fx(and)0 2440 y(natural)31 b(isomorphisms)g Ft(')d Fx(:)f Fu(G)6 b(F)1271 2412 y(\030)1272 2444 y Fx(=)1376 2440 y(Id)1465 2455 y Fr(C)1543 2440 y Fx(and)33 b Ft( )e Fx(:)d Fu(F)10 b(G)2056 2412 y(\030)2057 2444 y Fx(=)2161 2440 y(Id)2251 2455 y Fr(D)2312 2440 y Fx(.)149 2556 y(A)32 b(con)m(tra)m(v)-5 b(arian)m(t)32 b(functor)g Fu(F)37 b Fx(:)28 b Fu(C)34 b(\000)-60 b(!)27 b(D)35 b Fx(is)c(called)g(a)g Fs(duality)k(of)f(c)-5 b(ate)g(gories)31 b Fx(if)g(there)h(exists)0 2672 y(a)42 b(con)m(tra)m(v)-5 b(arian)m(t)41 b(functor)h Fu(G)50 b Fx(:)44 b Fu(D)i(\000)-60 b(!)44 b(C)k Fx(and)43 b(natural)d(isomorphisms)g Ft(')k Fx(:)g Fu(G)6 b(F)3145 2645 y(\030)3146 2677 y Fx(=)3266 2672 y(Id)3356 2687 y Fr(C)3443 2672 y Fx(and)0 2789 y Ft( )32 b Fx(:)27 b Fu(F)10 b(G)324 2761 y(\030)325 2793 y Fx(=)429 2789 y(Id)518 2804 y Fr(D)579 2789 y Fx(.)149 2905 y(A)29 b(category)f Fu(C)34 b Fx(is)28 b(said)f(to)h(b)s(e)g Fs(e)-5 b(quivalent)28 b Fx(to)f(a)h(category)h Fu(D)h Fx(if)d(there)i(exists)g(an)f(equiv)-5 b(alence)0 3021 y Fu(F)38 b Fx(:)29 b Fu(C)34 b(\000)-60 b(!)29 b(D)s Fx(.)44 b(A)34 b(category)f Fu(C)39 b Fx(is)33 b(said)f(to)h(b)s(e)g Fs(dual)g Fx(to)g(a)g(category)g Fu(D)j Fx(if)c(there)h(exists)h(a)f(dualit)m(y)0 3137 y Fu(F)k Fx(:)28 b Fu(C)34 b(\000)-60 b(!)28 b(D)s Fx(.)149 3312 y Fv(Problem)36 b(8.3.3.)49 b Fx(1.)44 b(Sho)m(w)33 b(that)f(the)h(dual)f(category)h Fu(C)2337 3276 y Fq(op)2444 3312 y Fx(is)f(dual)g(to)g(the)h(category)g Fu(C)6 b Fx(.)149 3428 y(2.)46 b(Let)33 b Fu(D)j Fx(b)s(e)e(a)f(category)g(dual) g(to)g(the)g(category)h Fu(C)6 b Fx(.)46 b(Sho)m(w)34 b(that)f Fu(D)j Fx(is)d(equiv)-5 b(alen)m(t)33 b(to)g(the)0 3544 y(dual)f(category)h Fu(C)668 3508 y Fq(op)742 3544 y Fx(.)149 3660 y(3.)52 b(Let)35 b Fu(F)42 b Fx(:)32 b Fu(C)38 b(\000)-60 b(!)33 b(D)k Fx(b)s(e)f(an)f(equiv)-5 b(alence)35 b(with)g(resp)s(ect)i(to)e Fu(G)j Fx(:)32 b Fu(D)j(\000)-60 b(!)32 b(C)6 b Fx(,)37 b Ft(')32 b Fx(:)g Fu(G)6 b(F)3329 3633 y(\030)3330 3665 y Fx(=)3438 3660 y(Id)3528 3675 y Fr(C)3573 3660 y Fx(,)0 3777 y(and)32 b Ft( )f Fx(:)d Fu(F)10 b(G)512 3749 y(\030)513 3781 y Fx(=)617 3777 y(Id)707 3792 y Fr(D)768 3777 y Fx(.)43 b(Sho)m(w)32 b(that)f Fu(G)j Fx(:)28 b Fu(D)i(\000)-60 b(!)28 b(C)37 b Fx(is)31 b(an)h(equiv)-5 b(alence.)43 b(Sho)m(w)32 b(that)f Fu(G)38 b Fx(is)31 b(uniquely)0 3893 y(determined)h(b)m(y)i Fu(F)42 b Fx(up)33 b(to)f(a)g(natural)g (isomorphism.)p eop %%Page: 10 10 10 9 bop 0 236 a Fo(10)1456 b(8.)77 b(TOOLBO)n(X)1305 425 y Fv(4.)94 b(T)-9 b(ensor)37 b(Pro)s(ducts)149 600 y(De\014nition)g(and)h(Remark)f(8.4.1.)49 b Fx(Let)40 b Ft(M)1900 615 y Fq(R)1997 600 y Fx(and)2193 615 y Fq(R)2251 600 y Ft(N)50 b Fx(b)s(e)39 b Ft(R)q Fx(-mo)s(dules,)h(and)f(let)f Ft(A)i Fx(b)s(e)0 716 y(an)32 b(ab)s(elian)f(group.)44 b(A)32 b(map)g Ft(f)38 b Fx(:)28 b Ft(M)33 b Fu(\002)23 b Ft(N)38 b Fu(\000)-60 b(!)28 b Ft(A)k Fx(is)g(called)g Ft(R)q Fs(-biline)-5 b(ar)31 b Fx(if)149 852 y(1.)42 b Ft(f)11 b Fx(\()p Ft(m)22 b Fx(+)g Ft(m)654 816 y Fr(0)678 852 y Ft(;)17 b(n)p Fx(\))27 b(=)h Ft(f)11 b Fx(\()p Ft(m;)17 b(n)p Fx(\))22 b(+)g Ft(f)11 b Fx(\()p Ft(m)1573 816 y Fr(0)1596 852 y Ft(;)17 b(n)p Fx(\))p Ft(;)149 969 y Fx(2.)42 b Ft(f)11 b Fx(\()p Ft(m;)17 b(n)22 b Fx(+)g Ft(n)729 932 y Fr(0)752 969 y Fx(\))28 b(=)g Ft(f)11 b Fx(\()p Ft(m;)17 b(n)p Fx(\))22 b(+)g Ft(f)11 b Fx(\()p Ft(m;)17 b(n)1648 932 y Fr(0)1671 969 y Fx(\))p Ft(;)149 1085 y Fx(3.)42 b Ft(f)11 b Fx(\()p Ft(mr)m(;)17 b(n)p Fx(\))28 b(=)f Ft(f)11 b Fx(\()p Ft(m;)17 b(r)s(n)p Fx(\))0 1221 y(for)32 b(all)f Ft(r)f Fu(2)e Ft(R)q(;)49 b(m;)17 b(m)818 1185 y Fr(0)870 1221 y Fu(2)28 b Ft(M)5 b(;)49 b(n;)17 b(n)1299 1185 y Fr(0)1350 1221 y Fu(2)28 b Ft(N)10 b Fx(.)149 1338 y(Let)33 b(Bil)447 1353 y Fq(R)505 1338 y Fx(\()p Ft(M)5 b(;)17 b(N)10 b Fx(;)17 b Ft(A)p Fx(\))33 b(denote)g(the)g(set)h(of)e(all)e Ft(R)q Fx(-bilinear)g(maps)i Ft(f)39 b Fx(:)27 b Ft(M)33 b Fu(\002)23 b Ft(N)38 b Fu(\000)-60 b(!)28 b Ft(A)p Fx(.)149 1454 y(Bil)273 1469 y Fq(R)330 1454 y Fx(\()p Ft(M)5 b(;)17 b(N)10 b Fx(;)17 b Ft(A)p Fx(\))33 b(is)f(an)h(ab)s(elian)e(group)h(with)g(\()p Ft(f)h Fx(+)22 b Ft(g)t Fx(\)\()p Ft(m;)17 b(n)p Fx(\))27 b(:=)h Ft(f)11 b Fx(\()p Ft(m;)17 b(n)p Fx(\))22 b(+)g Ft(g)t Fx(\()p Ft(m;)17 b(n)p Fx(\).)149 1621 y Fv(De\014nition)37 b(8.4.2.)49 b Fx(Let)39 b Ft(M)1258 1636 y Fq(R)1355 1621 y Fx(and)1551 1636 y Fq(R)1609 1621 y Ft(N)50 b Fx(b)s(e)39 b Ft(R)q Fx(-mo)s(dules.)62 b(An)39 b(ab)s(elian)e(group)i Ft(M)f Fu(\012)3427 1636 y Fq(R)3511 1621 y Ft(N)0 1737 y Fx(together)33 b(with)f(an)g Ft(R)q Fx(-bilinear)e(map)913 1894 y Fu(\012)e Fx(:)g Ft(M)k Fu(\002)23 b Ft(N)38 b Fu(3)28 b Fx(\()p Ft(m;)17 b(n)p Fx(\))28 b Fu(7!)g Ft(m)22 b Fu(\012)h Ft(n)k Fu(2)i Ft(M)k Fu(\012)2519 1909 y Fq(R)2599 1894 y Ft(N)0 2051 y Fx(is)j(called)e(a)i Fs(tensor)i(pr)-5 b(o)g(duct)38 b(of)74 b Ft(M)47 b Fs(and)73 b Ft(N)47 b Fs(over)36 b Ft(R)h Fx(if)e(for)g(eac)m(h)i(ab)s(elian)d(group)i Ft(A)g Fx(and)g(for)0 2167 y(eac)m(h)k Ft(R)q Fx(-bilinear)d(map)i Ft(f)50 b Fx(:)40 b Ft(M)e Fu(\002)27 b Ft(N)50 b Fu(\000)-60 b(!)40 b Ft(A)f Fx(there)i(exists)f(a)f(unique)h(group)g(homomorphism)0 2284 y Ft(g)31 b Fx(:)d Ft(M)33 b Fu(\012)337 2299 y Fq(R)417 2284 y Ft(N)38 b Fu(\000)-60 b(!)28 b Ft(A)33 b Fx(suc)m(h)h(that)e(the)h(diagram)1384 2464 y Ft(M)g Fu(\002)23 b Ft(N)154 b(M)33 b Fu(\012)2047 2479 y Fq(R)2127 2464 y Ft(N)p 1728 2439 86 4 v 1731 2437 a Fl(-)1732 2410 y Fu(\012)1666 2706 y Ft(f)1610 2589 y Fl(@)1693 2672 y(@)1776 2755 y(@)1859 2838 y(@)1878 2857 y(@)-83 b(R)1993 2959 y Ft(A)p 2027 2857 4 351 v 2029 2857 a Fl(?)2068 2693 y Ft(g)0 3082 y Fx(comm)m(utes.)71 b(The)43 b(elemen)m(ts)f(of)f Ft(M)e Fu(\012)1472 3097 y Fq(R)1559 3082 y Ft(N)52 b Fx(are)41 b(called)g Fs(tensors)p Fx(,)j(the)e(elemen) m(ts)g(of)f(the)h(form)0 3198 y Ft(m)22 b Fu(\012)h Ft(n)33 b Fx(are)f(called)g Fs(de)-5 b(c)g(omp)g(osable)33 b(tensors)p Fx(.)149 3315 y Fs(Warning:)42 b Fx(If)29 b(y)m(ou)i(w)m(an)m(t)f(to)f (de\014ne)i(a)f(homomorphism)d Ft(f)38 b Fx(:)28 b Ft(M)f Fu(\012)2604 3330 y Fq(R)2678 3315 y Ft(N)38 b Fu(\000)-60 b(!)28 b Ft(A)i Fx(with)f(a)g(tensor)0 3431 y(pro)s(duct)f(as)g(domain) e(y)m(ou)j Fs(must)f Fx(de\014ne)h(it)e(b)m(y)i(giving)d(an)i Ft(R)q Fx(-bilinear)d(map)j(de\014ned)h(on)f Ft(M)23 b Fu(\002)13 b Ft(N)d Fx(.)149 3598 y Fv(Lemma)38 b(8.4.3.)49 b Fs(A)35 b(tensor)f(pr)-5 b(o)g(duct)35 b Fx(\()p Ft(M)e Fu(\012)1848 3613 y Fq(R)1928 3598 y Ft(N)5 b(;)17 b Fu(\012)p Fx(\))34 b Fs(de\014ne)-5 b(d)34 b(by)h Ft(M)2754 3613 y Fq(R)2847 3598 y Fs(and)3036 3613 y Fq(R)3093 3598 y Ft(N)46 b Fs(is)34 b(unique)0 3714 y(up)h(to)g(a)g(unique)g (isomorphism.)149 3882 y Fi(Pr)n(oof.)42 b Fx(Let)33 b(\()p Ft(M)g Fu(\012)951 3897 y Fq(R)1031 3882 y Ft(N)5 b(;)17 b Fu(\012)p Fx(\))33 b(and)g(\()p Ft(M)g Fg(\002)1738 3897 y Fq(R)1818 3882 y Ft(N)5 b(;)17 b Fg(\002)p Fx(\))33 b(b)s(e)g(tensor)g(pro)s(ducts.)44 b(Then)1642 4023 y Ft(M)33 b Fu(\002)23 b Ft(N)1173 4262 y Fu(\012)1580 4103 y Fl(\010)1497 4145 y(\010)1414 4186 y(\010)1331 4228 y(\010)1248 4269 y(\010)1165 4311 y(\010)1082 4352 y(\010)999 4394 y(\010)962 4413 y(\010)-83 b(\031)1617 4271 y Fg(\002)1649 4145 y Fl(\000)1566 4228 y(\000)1483 4311 y(\000)1400 4394 y(\000)1381 4413 y(\000)g(\011)1906 4262 y Fu(\012)1868 4145 y Fl(@)1951 4228 y(@)2034 4311 y(@)2117 4394 y(@)2136 4413 y(@)g(R)2350 4271 y Fg(\002)1936 4103 y Fl(H)2020 4145 y(H)2103 4186 y(H)2186 4228 y(H)2269 4269 y(H)2352 4311 y(H)2435 4352 y(H)2518 4394 y(H)2555 4413 y(H)g(j)639 4507 y Ft(M)33 b Fu(\012)843 4522 y Fq(R)923 4507 y Ft(N)125 b(M)33 b Fg(\002)1330 4522 y Fq(R)1410 4507 y Ft(N)p 1041 4483 57 4 v 1015 4481 a Fl(-)1041 4462 y Ft(h)p 1528 4483 545 4 v 1990 4481 a Fl(-)1773 4462 y Ft(k)2101 4507 y(M)g Fu(\012)2305 4522 y Fq(R)2385 4507 y Ft(N)126 b(M)32 b Fg(\002)2792 4522 y Fq(R)2873 4507 y Ft(N)p 2503 4483 57 4 v 2477 4481 a Fl(-)2503 4462 y Ft(h)0 4650 y Fx(implies)e Ft(k)h Fx(=)c Ft(h)572 4614 y Fr(\000)p Fn(1)667 4650 y Fx(.)p 3529 4650 4 66 v 3533 4587 59 4 v 3533 4650 V 3591 4650 4 66 v 149 4816 a(Because)k(of)f(this)f(fact)g(w)m(e)i(will)c (henceforth)j(talk)f(ab)s(out)g Fs(the)h Fx(tensor)g(pro)s(duct)g(of)f Ft(M)40 b Fx(and)29 b Ft(N)0 4932 y Fx(o)m(v)m(er)34 b Ft(R)q Fx(.)149 5099 y Fv(Prop)s(osition)i(8.4.4.)49 b Fx(\(Rules)26 b(of)g(computation)e(in)h(a)h(tensor)h(pro)s(duct\))i Fs(L)-5 b(et)29 b Fx(\()p Ft(M)20 b Fu(\012)3291 5114 y Fq(R)3358 5099 y Ft(N)5 b(;)17 b Fu(\012)p Fx(\))0 5216 y Fs(b)-5 b(e)35 b(the)f(tensor)h(pr)-5 b(o)g(duct.)45 b(Then)34 b(we)g(have)h(for)f(al)5 b(l)35 b Ft(r)30 b Fu(2)e Ft(R)q Fs(,)35 b Ft(m;)17 b(m)2404 5179 y Fr(0)2455 5216 y Fu(2)28 b Ft(M)10 b Fs(,)36 b Ft(n;)17 b(n)2879 5179 y Fr(0)2930 5216 y Fu(2)28 b Ft(N)p eop %%Page: 11 11 11 10 bop 1348 236 a Fo(4.)77 b(TENSOR)32 b(PR)n(ODUCTS)1245 b(11)149 425 y Fx(1.)42 b Ft(M)33 b Fu(\012)471 440 y Fq(R)551 425 y Ft(N)38 b Fx(=)28 b Fu(f)821 350 y Ff(P)926 454 y Fq(i)971 425 y Ft(m)1056 440 y Fq(i)1106 425 y Fu(\012)23 b Ft(n)1264 440 y Fq(i)1327 425 y Fu(j)35 b Ft(m)1475 440 y Fq(i)1531 425 y Fu(2)28 b Ft(M)5 b(;)17 b(n)1826 440 y Fq(i)1882 425 y Fu(2)28 b Ft(N)10 b Fu(g)p Ft(;)149 541 y Fx(2.)42 b(\()p Ft(m)22 b Fx(+)g Ft(m)595 505 y Fr(0)619 541 y Fx(\))g Fu(\012)h Ft(n)k Fx(=)h Ft(m)22 b Fu(\012)h Ft(n)f Fx(+)g Ft(m)1438 505 y Fr(0)1484 541 y Fu(\012)h Ft(n;)149 658 y Fx(3.)42 b Ft(m)22 b Fu(\012)h Fx(\()p Ft(n)f Fx(+)g Ft(n)748 622 y Fr(0)772 658 y Fx(\))27 b(=)h Ft(m)22 b Fu(\012)h Ft(n)f Fx(+)g Ft(m)h Fu(\012)f Ft(n)1591 622 y Fr(0)1615 658 y Ft(;)149 774 y Fx(4.)42 b Ft(mr)29 b Fu(\012)d Ft(n)37 b Fx(=)g Ft(m)26 b Fu(\012)h Ft(r)s(n)40 b Fs(\(observe)e(in)i(p)-5 b(articular,)41 b(that)f Fu(\012)e Fx(:)f Ft(M)f Fu(\002)27 b Ft(N)47 b Fu(\000)-58 b(!)38 b Ft(M)f Fu(\012)26 b Ft(N)50 b Fs(is)40 b(not)267 890 y(inje)-5 b(ctive)34 b(in)g(gener)-5 b(al\),)149 1006 y Fx(5.)42 b Fs(if)35 b Ft(f)40 b Fx(:)29 b Ft(M)34 b Fu(\002)23 b Ft(N)39 b Fu(\000)-58 b(!)30 b Ft(A)36 b Fs(is)f(an)g Ft(R)q Fs(-biline)-5 b(ar)35 b(map)g(and)g Ft(g)d Fx(:)d Ft(M)34 b Fu(\012)2560 1021 y Fq(R)2641 1006 y Ft(N)39 b Fu(\000)-58 b(!)30 b Ft(A)36 b Fs(is)f(the)h(induc)-5 b(e)g(d)267 1123 y(homomorphism,)32 b(then)1364 1239 y Ft(g)t Fx(\()p Ft(m)22 b Fu(\012)g Ft(n)p Fx(\))28 b(=)g Ft(f)11 b Fx(\()p Ft(m;)17 b(n)p Fx(\))p Ft(:)149 1414 y Fi(Pr)n(oof.)42 b Fx(1.)83 b(Let)46 b Ft(B)56 b Fx(:=)50 b Fu(h)p Ft(m)31 b Fu(\012)h Ft(n)p Fu(i)50 b(\022)h Ft(M)42 b Fu(\012)1916 1429 y Fq(R)2005 1414 y Ft(N)56 b Fx(denote)47 b(the)f(subgroup)h(of)e Ft(M)d Fu(\012)3422 1429 y Fq(R)3511 1414 y Ft(N)0 1530 y Fx(generated)24 b(b)m(y)h(the)f(decomp)s(osable)f(tensors)i Ft(m)t Fu(\012)t Ft(n)p Fx(.)40 b(Let)24 b Ft(j)34 b Fx(:)28 b Ft(B)k Fu(\000)-60 b(!)28 b Ft(M)14 b Fu(\012)2682 1545 y Fq(R)2745 1530 y Ft(N)34 b Fx(b)s(e)23 b(the)h(em)m(b)s(edding)0 1646 y(homomorphism.)40 b(W)-8 b(e)29 b(get)h(an)f(induced)h(map)e Fu(\012)1832 1610 y Fr(0)1884 1646 y Fx(:)f Ft(M)g Fu(\002)15 b Ft(N)39 b Fu(\000)-60 b(!)28 b Ft(B)5 b Fx(.)42 b(In)30 b(the)g(follo)m(wing)c(diagram)1140 1848 y Ft(M)33 b Fu(\002)23 b Ft(N)301 b(B)p 1484 1820 234 4 v 1635 1818 a Fl(-)1550 1791 y Fu(\012)1627 1755 y Fr(0)2086 1844 y Ft(M)33 b Fu(\012)2290 1859 y Fq(R)2371 1844 y Ft(N)p 1854 1820 204 4 v 1975 1818 a Fl(-)1932 1779 y Ft(j)1746 2332 y(B)266 b(M)33 b Fu(\012)2290 2347 y Fq(R)2371 2332 y Ft(N)p 1854 2307 V 1975 2305 a Fl(-)1932 2267 y Ft(j)1380 2094 y Fu(\012)1457 2058 y Fr(0)1366 1969 y Fl(@)1449 2052 y(@)1532 2135 y(@)1615 2218 y(@)1634 2237 y(@)-83 b(R)p 1784 2237 4 351 v 68 w(?)1604 2089 y Fx(id)1686 2104 y Fq(B)p 2271 2237 V 2273 2237 a Fl(?)2312 2085 y Ft(j)6 b(p)2090 2073 y(p)2122 1969 y Fl(\000)2039 2052 y(\000)1956 2135 y(\000)1873 2218 y(\000)1854 2237 y(\000)-83 b(\011)0 2469 y Fx(w)m(e)39 b(ha)m(v)m(e)g(id)461 2484 y Fq(B)538 2469 y Fu(\016\012)665 2432 y Fr(0)725 2469 y Fx(=)e Fu(\012)915 2432 y Fr(0)939 2469 y Fx(,)i Ft(p)f Fx(with)g Ft(p)26 b Fu(\016)f Ft(j)32 b Fu(\016)26 b(\012)1695 2432 y Fr(0)1755 2469 y Fx(=)37 b Ft(p)26 b Fu(\016)f(\012)38 b Fx(=)f Fu(\012)2323 2432 y Fr(0)2384 2469 y Fx(exists)i(since)f Fu(\012)2980 2432 y Fr(0)3042 2469 y Fx(is)g Ft(R)q Fx(-bilinear.)0 2585 y(Because)47 b(of)e Ft(j)6 b(p)31 b Fu(\016)g(\012)50 b Fx(=)g Ft(j)36 b Fu(\016)31 b(\012)1204 2549 y Fr(0)1278 2585 y Fx(=)49 b Fu(\012)h Fx(=)g(id)1737 2600 y Fq(M)7 b Fr(\012)1867 2611 y Fk(R)1917 2600 y Fq(N)2001 2585 y Fu(\016\012)46 b Fx(w)m(e)h(get)e Ft(j)6 b(p)50 b Fx(=)f(id)2858 2600 y Fq(M)7 b Fr(\012)2988 2611 y Fk(R)3038 2600 y Fq(N)3105 2585 y Fx(,)49 b(hence)e(the)0 2701 y(em)m(b)s(edding)32 b Ft(j)39 b Fx(is)32 b(surjectiv)m(e)i(and)e(th)m(us)i(the)f(iden)m (tit)m(y)-8 b(.)149 2817 y(2.)44 b(\()p Ft(m)22 b Fx(+)g Ft(m)597 2781 y Fr(0)621 2817 y Fx(\))g Fu(\012)g Ft(n)28 b Fx(=)g Fu(\012)p Fx(\()p Ft(m)23 b Fx(+)f Ft(m)1376 2781 y Fr(0)1399 2817 y Ft(;)17 b(n)p Fx(\))28 b(=)f Fu(\012)p Fx(\()p Ft(m;)17 b(n)p Fx(\))23 b(+)f Fu(\012)p Fx(\()p Ft(m)2331 2781 y Fr(0)2355 2817 y Ft(;)17 b(n)p Fx(\))28 b(=)f Ft(m)c Fu(\012)f Ft(n)h Fx(+)f Ft(m)3097 2781 y Fr(0)3143 2817 y Fu(\012)g Ft(n)p Fx(.)149 2933 y(3.)44 b(and)32 b(4.)44 b(analogously)-8 b(.)149 3050 y(5.)44 b(is)32 b(precisely)h(the)g(de\014nition)e(of)h(the)h(induced)g (homomorphism.)p 3529 3050 4 66 v 3533 2987 59 4 v 3533 3050 V 3591 3050 4 66 v 149 3224 a Fv(Remark)k(8.4.5.)50 b Fx(T)-8 b(o)25 b(construct)h(tensor)f(pro)s(ducts,)i(w)m(e)f(use)g (the)f(notion)f(of)g(a)h(free)g(mo)s(dule.)149 3340 y(Let)32 b Ft(X)38 b Fx(b)s(e)32 b(a)e(set)i(and)f Ft(R)h Fx(b)s(e)g(a)e(ring.) 42 b(An)32 b Ft(R)q Fx(-mo)s(dule)d Ft(R)q(X)39 b Fx(together)31 b(with)g(a)f(map)h Ft(\023)d Fx(:)f Ft(X)36 b Fu(\000)-60 b(!)0 3457 y Ft(R)q(X)39 b Fx(is)30 b(called)g(a)g Fs(fr)-5 b(e)g(e)30 b Ft(R)q Fx(-)p Fs(mo)-5 b(dule)33 b(gener)-5 b(ate)g(d)32 b(by)f Ft(X)8 b Fx(,)31 b(if)f(for)g(ev)m(ery)j Ft(R)q Fx(-mo)s(dule)28 b Ft(M)42 b Fx(and)31 b(for)f(ev)m(ery)0 3573 y(map)35 b Ft(f)43 b Fx(:)32 b Ft(X)40 b Fu(\000)-60 b(!)33 b Ft(M)46 b Fx(there)36 b(exists)g(a)f(unique)h(homomorphism)d (of)h Ft(R)q Fx(-mo)s(dules)g Ft(g)i Fx(:)d Ft(R)q(X)40 b Fu(\000)-60 b(!)32 b Ft(M)0 3689 y Fx(suc)m(h)i(that)e(the)h(diagram) 1489 3811 y Ft(X)369 b(R)q(X)p 1607 3779 304 4 v 1828 3777 a Fl(-)1741 3759 y Ft(\023)1658 4046 y(f)1602 3929 y Fl(@)1685 4012 y(@)1768 4095 y(@)1851 4178 y(@)1870 4197 y(@)-83 b(R)1969 4299 y Ft(M)p 2020 4197 4 351 v 2021 4197 a Fl(?)2060 4033 y Ft(g)0 4402 y Fx(comm)m(utes.)149 4518 y(F)-8 b(ree)34 b Ft(R)q Fx(-mo)s(dules)f(exist)h(and)g(can)g(b)s (e)g(constructed)h(as)f Ft(R)q(X)k Fx(:=)30 b Fu(f)p Ft(\013)g Fx(:)g Ft(X)37 b Fu(\000)-60 b(!)30 b Ft(R)q Fu(j)j Fx(for)h(almost)0 4634 y(all)c Ft(x)f Fu(2)f Ft(X)35 b Fx(:)28 b Ft(\013)q Fx(\()p Ft(x)p Fx(\))g(=)f(0)p Fu(g)p Fx(.)149 4809 y Fv(Prop)s(osition)36 b(8.4.6.)49 b Fs(Given)40 b Ft(R)q Fs(-mo)-5 b(dules)40 b Ft(M)1936 4824 y Fq(R)2035 4809 y Fs(and)2229 4824 y Fq(R)2287 4809 y Ft(N)10 b Fs(.)62 b(Then)40 b(ther)-5 b(e)40 b(exists)g(a)g (tensor)0 4925 y(pr)-5 b(o)g(duct)35 b Fx(\()p Ft(M)e Fu(\012)587 4940 y Fq(R)667 4925 y Ft(N)5 b(;)17 b Fu(\012)p Fx(\))p Fs(.)149 5099 y Fi(Pr)n(oof.)42 b Fx(De\014ne)35 b Ft(M)f Fu(\012)1043 5114 y Fq(R)1124 5099 y Ft(N)41 b Fx(:=)30 b Fp(Z)p Fu(f)p Ft(M)h Fu(\002)24 b Ft(N)10 b Fu(g)p Ft(=U)44 b Fx(where)36 b Fp(Z)p Fu(f)p Ft(M)31 b Fu(\002)23 b Ft(N)10 b Fu(g)35 b Fx(is)e(a)h(free)g Fp(Z)p Fx(-mo)s(dule)0 5216 y(o)m(v)m(er)g Ft(M)f Fu(\002)22 b Ft(N)43 b Fx(\(the)33 b(free)g(ab)s(elian)e(group\))h(and)h Ft(U)43 b Fx(is)32 b(generated)i(b)m(y)p eop %%Page: 12 12 12 11 bop 0 236 a Fo(12)1456 b(8.)77 b(TOOLBO)n(X)267 425 y Ft(\023)p Fx(\()p Ft(m)23 b Fx(+)f Ft(m)630 389 y Fr(0)653 425 y Ft(;)17 b(n)p Fx(\))22 b Fu(\000)h Ft(\023)p Fx(\()p Ft(m;)17 b(n)p Fx(\))23 b Fu(\000)f Ft(\023)p Fx(\()p Ft(m)1491 389 y Fr(0)1515 425 y Ft(;)17 b(n)p Fx(\))267 541 y Ft(\023)p Fx(\()p Ft(m;)g(m)23 b Fx(+)f Ft(n)732 505 y Fr(0)755 541 y Fx(\))g Fu(\000)h Ft(\023)p Fx(\()p Ft(m;)17 b(n)p Fx(\))23 b Fu(\000)f Ft(\023)p Fx(\()p Ft(m;)17 b(n)1593 505 y Fr(0)1617 541 y Fx(\))267 658 y Ft(\023)p Fx(\()p Ft(mr)m(;)g(n)p Fx(\))23 b Fu(\000)f Ft(\023)p Fx(\()p Ft(m;)17 b(r)s(n)p Fx(\))0 794 y(for)32 b(all)f Ft(r)f Fu(2)e Ft(R)q Fx(,)33 b Ft(m;)17 b(m)802 758 y Fr(0)853 794 y Fu(2)28 b Ft(M)10 b Fx(,)33 b Ft(n;)17 b(n)1271 758 y Fr(0)1323 794 y Fu(2)28 b Ft(N)10 b Fx(.)44 b(Consider)537 948 y Ft(M)33 b Fu(\002)23 b Ft(N)344 b Fp(Z)p Fu(f)p Ft(M)30 b Fu(\002)22 b Ft(N)10 b Fu(g)p 881 925 276 4 v 1074 923 a Fl(-)1001 905 y Ft(\023)1971 948 y(M)33 b Fu(\012)2175 963 y Fq(R)2255 948 y Ft(N)p 1695 925 247 4 v 1859 923 a Fl(-)1791 905 y Ft(\027)2352 948 y Fx(=)28 b Fp(Z)p Fu(f)p Ft(M)i Fu(\002)23 b Ft(N)10 b Fu(g)p Ft(=U)2121 1445 y(A)1295 1192 y( )900 1019 y Fl(P)983 1047 y(P)1066 1074 y(P)1149 1102 y(P)1232 1130 y(P)1315 1157 y(P)1398 1185 y(P)1481 1213 y(P)1564 1240 y(P)1647 1268 y(P)1730 1296 y(P)1813 1323 y(P)1870 1342 y(P)-83 b(q)1853 1179 y Ft(\032)1528 1047 y Fl(Q)1612 1102 y(Q)1695 1157 y(Q)1778 1213 y(Q)1861 1268 y(Q)1944 1323 y(Q)1972 1342 y(Q)g(s)p 2156 1342 4 351 v 102 w(?)2196 1179 y Ft(g)0 1567 y Fx(Let)36 b Ft( )k Fx(b)s(e)c(giv)m(en.)54 b(Then)38 b(there)e(is)g(a)g(unique)g Ft(\032)e Fu(2)g Fx(Hom)o(\()p Fp(Z)p Fu(f)p Ft(M)f Fu(\002)25 b Ft(N)10 b Fu(g)p Ft(;)17 b(A)p Fx(\))36 b(suc)m(h)h(that)f Ft(\032\023)e Fx(=)g Ft( )t Fx(.)0 1684 y(Since)41 b Ft( )j Fx(is)c Ft(R)q Fx(-bilinear)d(w)m(e)42 b(get)e Ft(\032)p Fx(\()p Ft(\023)p Fx(\()p Ft(m)29 b Fx(+)e Ft(m)1727 1647 y Fr(0)1750 1684 y Ft(;)17 b(n)p Fx(\))28 b Fu(\000)g Ft(\023)p Fx(\()p Ft(m;)17 b(n)p Fx(\))28 b Fu(\000)g Ft(\023)p Fx(\()p Ft(m)2610 1647 y Fr(0)2634 1684 y Ft(n)p Fx(\)\))41 b(=)g Ft( )t Fx(\()p Ft(m)27 b Fx(+)h Ft(m)3332 1647 y Fr(0)3355 1684 y Ft(;)17 b(n)p Fx(\))28 b Fu(\000)0 1800 y Ft( )t Fx(\()p Ft(m;)17 b(n)p Fx(\))28 b Fu(\000)g Ft( )t Fx(\()p Ft(m)653 1764 y Fr(0)677 1800 y Ft(;)17 b(n)p Fx(\))42 b(=)g(0)e(and)h(similarly)d Ft(\032)p Fx(\()p Ft(\023)p Fx(\()p Ft(m;)17 b(n)28 b Fx(+)g Ft(n)2209 1764 y Fr(0)2233 1800 y Fx(\))f Fu(\000)i Ft(\023)p Fx(\()p Ft(m;)17 b(n)p Fx(\))28 b Fu(\000)h Ft(\023)p Fx(\()p Ft(m;)17 b(n)3094 1764 y Fr(0)3117 1800 y Fx(\)\))42 b(=)g(0)f(and)0 1916 y Ft(\032)p Fx(\()p Ft(\023)p Fx(\()p Ft(mr)m(;)17 b(n)p Fx(\))25 b Fu(\000)f Ft(\023)p Fx(\()p Ft(m;)17 b(r)s(n)p Fx(\)\))31 b(=)g(0.)49 b(So)34 b(w)m(e)i(get)e Ft(\032)p Fx(\()p Ft(U)10 b Fx(\))32 b(=)f(0.)49 b(This)35 b(implies)d(that)i (there)h(is)f(a)g(unique)0 2032 y Ft(g)j Fu(2)d Fx(Hom\()p Ft(M)h Fu(\012)632 2047 y Fq(R)715 2032 y Ft(N)5 b(;)17 b(A)p Fx(\))36 b(suc)m(h)h(that)f Ft(g)t(\027)k Fx(=)34 b Ft(\032)i Fx(\(homomorphism)d(theorem\).)54 b(Let)37 b Fu(\012)d Fx(:=)g Ft(\027)d Fu(\016)24 b Ft(\023)p Fx(.)0 2149 y(Then)47 b Fu(\012)g Fx(is)f(bilinear)e(since)j(\()p Ft(m)31 b Fx(+)h Ft(m)1470 2112 y Fr(0)1493 2149 y Fx(\))g Fu(\012)g Ft(n)51 b Fx(=)g Ft(\027)38 b Fu(\016)31 b Ft(\023)p Fx(\()p Ft(m)h Fx(+)f Ft(m)2456 2112 y Fr(0)2480 2149 y Ft(;)17 b(n)p Fx(\))51 b(=)g Ft(\027)6 b Fx(\()p Ft(\023)p Fx(\()p Ft(m)32 b Fx(+)g Ft(m)3272 2112 y Fr(0)3295 2149 y Ft(;)17 b(n)p Fx(\)\))51 b(=)0 2265 y Ft(\027)6 b Fx(\()p Ft(\023)p Fx(\()p Ft(m)29 b Fx(+)f Ft(m)467 2229 y Fr(0)491 2265 y Ft(;)17 b(n)p Fx(\))28 b Fu(\000)h Ft(\023)p Fx(\()p Ft(m;)17 b(n)p Fx(\))29 b Fu(\000)f Ft(\023)p Fx(\()p Ft(m)1353 2229 y Fr(0)1377 2265 y Ft(;)17 b(n)p Fx(\))29 b(+)f Ft(\023)p Fx(\()p Ft(m;)17 b(n)p Fx(\))28 b(+)h Ft(\023)p Fx(\()p Ft(m)2237 2229 y Fr(0)2261 2265 y Ft(;)17 b(n)p Fx(\)\))42 b(=)h Ft(\027)6 b Fx(\()p Ft(\023)p Fx(\()p Ft(m;)17 b(n)p Fx(\))29 b(+)f Ft(\023)p Fx(\()p Ft(m)3279 2229 y Fr(0)3303 2265 y Ft(;)17 b(n)p Fx(\)\))43 b(=)0 2381 y Ft(\027)25 b Fu(\016)18 b Ft(\023)p Fx(\()p Ft(m;)f(n)p Fx(\))j(+)e Ft(\027)25 b Fu(\016)18 b Ft(\023)p Fx(\()p Ft(m)850 2345 y Fr(0)874 2381 y Ft(;)f(n)p Fx(\))28 b(=)f Ft(m)19 b Fu(\012)g Ft(n)g Fx(+)f Ft(m)1601 2345 y Fr(0)1644 2381 y Fu(\012)h Ft(n)p Fx(.)43 b(The)31 b(other)g(t)m(w)m(o)h(prop)s(erties)f(are)f(obtained)h(in)0 2497 y(an)h(analogous)g(w)m(a)m(y)-8 b(.)149 2613 y(W)g(e)28 b(ha)m(v)m(e)g(to)f(sho)m(w)h(that)f(\()p Ft(M)21 b Fu(\012)1317 2628 y Fq(R)1386 2613 y Ft(N)5 b(;)17 b Fu(\012)p Fx(\))27 b(is)g(a)f(tensor)i(pro)s(duct.)42 b(The)28 b(ab)s(o)m(v)m(e)g(diagram) c(sho)m(ws)0 2730 y(that)38 b(for)f(eac)m(h)i(ab)s(elian)c(group)j Ft(A)g Fx(and)g(for)f(eac)m(h)i Ft(R)q Fx(-bilinear)c(map)i Ft( )j Fx(:)d Ft(M)g Fu(\002)26 b Ft(N)47 b Fu(\000)-60 b(!)36 b Ft(A)i Fx(there)0 2846 y(is)e(a)f Ft(g)i Fu(2)d Fx(Hom\()p Ft(M)h Fu(\012)818 2861 y Fq(R)900 2846 y Ft(N)5 b(;)17 b(A)p Fx(\))36 b(suc)m(h)i(that)e Ft(g)28 b Fu(\016)c(\012)34 b Fx(=)g Ft( )t Fx(.)53 b(Giv)m(en)36 b Ft(h)e Fu(2)g Fx(Hom)o(\()p Ft(M)i Fu(\012)3054 2861 y Fq(R)3136 2846 y Ft(N)5 b(;)17 b(A)p Fx(\))36 b(with)0 2962 y Ft(h)22 b Fu(\016)g(\012)28 b Fx(=)g Ft( )t Fx(.)43 b(Then)34 b Ft(h)22 b Fu(\016)g Ft(\027)29 b Fu(\016)22 b Ft(\023)28 b Fx(=)g Ft( )t Fx(.)43 b(This)33 b(implies)d Ft(h)22 b Fu(\016)g Ft(\027)34 b Fx(=)28 b Ft(\032)g Fx(=)f Ft(g)f Fu(\016)c Ft(\027)39 b Fx(hence)34 b Ft(g)d Fx(=)c Ft(h)p Fx(.)p 3529 2962 4 66 v 3533 2900 59 4 v 3533 2962 V 3591 2962 4 66 v 149 3127 a Fv(Prop)s(osition)36 b(and)i(De\014nition)e(8.4.7.)50 b Fs(Given)34 b(two)h(homomorphisms) 836 3284 y Ft(f)j Fu(2)28 b Fx(Hom)1236 3299 y Fq(R)1294 3284 y Fx(\()p Ft(M)5 b(:;)17 b(M)1606 3243 y Fr(0)1630 3284 y Ft(:)p Fx(\))35 b Fs(and)f Ft(g)d Fu(2)d Fx(Hom)2311 3299 y Fq(R)2369 3284 y Fx(\()p Ft(:N)5 b(;)17 b(:N)2676 3243 y Fr(0)2699 3284 y Fx(\))p Ft(:)0 3441 y Fs(Then)34 b(ther)-5 b(e)35 b(is)g(a)f(unique)h(homomorphism)1040 3598 y Ft(f)e Fu(\012)1198 3613 y Fq(R)1278 3598 y Ft(g)e Fu(2)d Fx(Hom\()p Ft(M)k Fu(\012)1895 3613 y Fq(R)1976 3598 y Ft(N)5 b(;)17 b(M)2207 3557 y Fr(0)2253 3598 y Fu(\012)2330 3613 y Fq(R)2410 3598 y Ft(N)2498 3557 y Fr(0)2522 3598 y Fx(\))0 3755 y Fs(such)35 b(that)g Ft(f)e Fu(\012)579 3770 y Fq(R)659 3755 y Ft(g)t Fx(\()p Ft(m)22 b Fu(\012)g Ft(n)p Fx(\))28 b(=)g Ft(f)11 b Fx(\()p Ft(m)p Fx(\))22 b Fu(\012)h Ft(g)t Fx(\()p Ft(n)p Fx(\))p Fs(,)34 b(i.e.)44 b(the)35 b(fol)5 b(lowing)33 b(diagr)-5 b(am)34 b(c)-5 b(ommutes)1298 4431 y Ft(M)1402 4395 y Fr(0)1448 4431 y Fu(\002)23 b Ft(N)1636 4395 y Fr(0)1854 4431 y Ft(M)1958 4395 y Fr(0)2004 4431 y Fu(\012)2081 4446 y Fq(R)2162 4431 y Ft(N)2250 4395 y Fr(0)p 1688 4404 137 4 v 1742 4402 a Fl(-)1718 4480 y Fu(\012)1321 3941 y Ft(M)33 b Fu(\002)23 b Ft(N)251 b(M)33 b Fu(\012)2081 3956 y Fq(R)2162 3941 y Ft(N)p 1665 3916 184 4 v 1766 3914 a Fl(-)1718 3887 y Fu(\012)p 1477 4334 4 351 v 1479 4334 a Fl(?)1209 4183 y Ft(f)g Fu(\002)22 b Ft(g)p 2062 4334 V 2064 4334 a Fl(?)2103 4183 y Ft(f)33 b Fu(\012)2261 4198 y Fq(R)2341 4183 y Ft(g)149 4620 y Fi(Pr)n(oof.)42 b Fu(\012)23 b(\016)f Fx(\()p Ft(f)33 b Fu(\002)22 b Ft(g)t Fx(\))32 b(is)g(bilinear.)p 3529 4620 4 66 v 3533 4558 59 4 v 3533 4620 V 3591 4620 4 66 v 149 4786 a Fv(Notation)37 b(8.4.8.)49 b Fx(W)-8 b(e)37 b(often)f(write)g Ft(f)g Fu(\012)1768 4801 y Fq(R)1850 4786 y Ft(N)45 b Fx(:=)34 b Ft(f)i Fu(\012)2271 4801 y Fq(R)2354 4786 y Fx(1)2403 4801 y Fq(N)2506 4786 y Fx(and)g Ft(M)g Fu(\012)2906 4801 y Fq(R)2989 4786 y Ft(g)h Fx(:=)d(1)3259 4801 y Fq(M)3363 4786 y Fu(\012)3440 4801 y Fq(R)3523 4786 y Ft(g)t Fx(.)0 4902 y(W)-8 b(e)33 b(ha)m(v)m(e)h(the)f(follo)m(wing)d (rule)i(of)g(computation:)594 5059 y Ft(f)h Fu(\012)752 5074 y Fq(R)832 5059 y Ft(g)e Fx(=)d(\()p Ft(f)33 b Fu(\012)1210 5074 y Fq(R)1290 5059 y Ft(N)1378 5018 y Fr(0)1402 5059 y Fx(\))22 b Fu(\016)g Fx(\()p Ft(M)33 b Fu(\012)1776 5074 y Fq(R)1856 5059 y Ft(g)t Fx(\))27 b(=)g(\()p Ft(M)2217 5018 y Fr(0)2264 5059 y Fu(\012)2341 5074 y Fq(R)2421 5059 y Ft(g)t Fx(\))21 b Fu(\016)h Fx(\()p Ft(f)33 b Fu(\012)2799 5074 y Fq(R)2879 5059 y Ft(N)10 b Fx(\))0 5216 y(since)33 b Ft(f)g Fu(\002)22 b Ft(g)31 b Fx(=)d(\()p Ft(f)33 b Fu(\002)22 b Ft(N)907 5179 y Fr(0)931 5216 y Fx(\))g Fu(\016)g Fx(\()p Ft(M)33 b Fu(\002)23 b Ft(g)t Fx(\))k(=)g(\()p Ft(M)1689 5179 y Fr(0)1735 5216 y Fu(\002)c Ft(g)t Fx(\))e Fu(\016)h Fx(\()p Ft(f)33 b Fu(\002)23 b Ft(N)10 b Fx(\).)p eop %%Page: 13 13 13 12 bop 1348 236 a Fo(4.)77 b(TENSOR)32 b(PR)n(ODUCTS)1245 b(13)149 425 y Fv(Prop)s(osition)36 b(8.4.9.)49 b Fs(The)35 b(fol)5 b(lowing)33 b(de\014ne)h(c)-5 b(ovariant)34 b(functors)149 562 y Fx(1.)42 b Fs(-)22 b Fu(\012)g Ft(N)39 b Fx(:)27 b Fv(Mo)s(d)q Fs(-)p Ft(R)i Fu(\000)-58 b(!)28 b Fv(Ab)p Fs(;)149 678 y Fx(2.)42 b Ft(M)33 b Fu(\012)22 b Fs(-)28 b Fx(:)g Ft(R)q Fs(-)p Fv(Mo)s(d)g Fu(\000)-58 b(!)28 b Fv(Ab)p Fs(;)149 794 y Fx(3.)42 b Fs(-)22 b Fu(\012)g Fs(-)28 b Fx(:)g Fv(Mo)s(d)q Fs(-)o Ft(R)c Fu(\002)e Ft(R)q Fs(-)p Fv(Mo)s(d)29 b Fu(\000)-58 b(!)28 b Fv(Ab)p Fs(.)149 956 y Fi(Pr)n(oof.)42 b Fx(\()p Ft(f)34 b Fu(\002)23 b Ft(g)t Fx(\))f Fu(\016)g Fx(\()p Ft(f)1034 919 y Fr(0)1080 956 y Fu(\002)h Ft(g)1231 919 y Fr(0)1254 956 y Fx(\))29 b(=)g Ft(f)11 b(f)1544 919 y Fr(0)1589 956 y Fu(\002)23 b Ft(g)t(g)1791 919 y Fr(0)1847 956 y Fx(implies)31 b(\()p Ft(f)i Fu(\012)2375 971 y Fq(R)2456 956 y Ft(g)t Fx(\))22 b Fu(\016)g Fx(\()p Ft(f)2736 919 y Fr(0)2782 956 y Fu(\012)2859 971 y Fq(R)2940 956 y Ft(g)2991 919 y Fr(0)3013 956 y Fx(\))29 b(=)g Ft(f)11 b(f)3303 919 y Fr(0)3349 956 y Fu(\002)23 b Ft(g)t(g)3551 919 y Fr(0)3573 956 y Fx(.)0 1072 y(F)-8 b(urthermore)32 b(1)614 1087 y Fq(M)715 1072 y Fu(\002)22 b Fx(1)863 1087 y Fq(N)958 1072 y Fx(=)28 b(1)1111 1087 y Fq(M)7 b Fr(\002)p Fq(N)1340 1072 y Fx(implies)30 b(1)1720 1087 y Fq(M)1821 1072 y Fu(\012)1898 1087 y Fq(R)1978 1072 y Fx(1)2027 1087 y Fq(N)2122 1072 y Fx(=)e(1)2275 1087 y Fq(M)7 b Fr(\012)2405 1098 y Fk(R)2454 1087 y Fq(N)2522 1072 y Fx(.)p 3529 1072 4 66 v 3533 1009 59 4 v 3533 1072 V 3591 1072 4 66 v 149 1233 a Fv(De\014nition)37 b(8.4.10.)49 b Fx(Let)24 b Ft(R)q Fx(,)i Ft(S)j Fx(b)s(e)c(rings)e(and) h(let)f Ft(M)35 b Fx(b)s(e)24 b(a)g(left)f Ft(R)q Fx(-mo)s(dule)f(and)i (a)g(righ)m(t)f Ft(S)6 b Fx(-)0 1350 y(mo)s(dule.)41 b Ft(M)e Fx(is)27 b(called)g(an)h Ft(R)q Fx(-)p Ft(S)6 b Fx(-)p Fs(bimo)-5 b(dule)27 b Fx(if)g(\()p Ft(r)s(m)p Fx(\))p Ft(s)g Fx(=)h Ft(r)s Fx(\()p Ft(ms)p Fx(\).)41 b(W)-8 b(e)29 b(de\014ne)h(Hom)3053 1365 y Fq(R)p Fx(-)p Fq(S)3190 1350 y Fx(\()p Ft(:M)5 b(:;)17 b(:N)5 b(:)p Fx(\))0 1466 y(:=)28 b(Hom)334 1481 y Fq(R)392 1466 y Fx(\()p Ft(:M)5 b(;)17 b(:N)10 b Fx(\))22 b Fu(\\)h Fx(Hom)1067 1481 y Fq(S)1118 1466 y Fx(\()p Ft(M)5 b(:;)17 b(N)5 b(:)p Fx(\).)149 1627 y Fv(Remark)37 b(8.4.11.)50 b Fx(Let)34 b Ft(M)1211 1642 y Fq(S)1297 1627 y Fx(b)s(e)g(a)g(righ)m(t)g Ft(S)6 b Fx(-mo)s(dule)32 b(and)j(let)e Ft(R)25 b Fu(\002)f Ft(M)41 b Fu(\000)-60 b(!)31 b Ft(M)45 b Fx(a)34 b(map.)48 b Ft(M)0 1743 y Fx(is)32 b(an)h Ft(R)q Fx(-)p Ft(S)6 b Fx(-bimo)s(dule)29 b(if)i(and)i(only)f(if)149 1880 y(1.)42 b Fu(8)p Ft(r)31 b Fu(2)d Ft(R)g Fx(:)g(\()p Ft(M)39 b Fu(3)28 b Ft(m)g Fu(7!)f Ft(r)s(m)h Fu(2)g Ft(M)10 b Fx(\))28 b Fu(2)g Fx(Hom)1874 1895 y Fq(S)1925 1880 y Fx(\()p Ft(M)5 b(:;)17 b(M)5 b(:)p Fx(\),)149 1996 y(2.)42 b Fu(8)p Ft(r)m(;)17 b(r)454 1960 y Fr(0)505 1996 y Fu(2)28 b Ft(R)q(;)17 b(m)28 b Fu(2)g Ft(M)38 b Fx(:)28 b(\()p Ft(r)d Fx(+)d Ft(r)1364 1960 y Fr(0)1387 1996 y Fx(\))p Ft(m)28 b Fx(=)f Ft(r)s(m)22 b Fx(+)g Ft(r)1940 1960 y Fr(0)1963 1996 y Ft(m)p Fx(,)149 2112 y(3.)42 b Fu(8)p Ft(r)m(;)17 b(r)454 2076 y Fr(0)505 2112 y Fu(2)28 b Ft(R)q(;)17 b(m)28 b Fu(2)g Ft(M)38 b Fx(:)28 b(\()p Ft(r)s(r)1244 2076 y Fr(0)1267 2112 y Fx(\))p Ft(m)f Fx(=)h Ft(r)s Fx(\()p Ft(r)1653 2076 y Fr(0)1676 2112 y Ft(m)p Fx(\),)149 2229 y(4.)42 b Fu(8)p Ft(m)28 b Fu(2)g Ft(M)39 b Fx(:)27 b(1)p Ft(m)h Fx(=)g Ft(m:)149 2390 y Fv(Lemma)38 b(8.4.12.)49 b Fs(L)-5 b(et)1071 2405 y Fq(R)1129 2390 y Ft(M)1223 2405 y Fq(S)1299 2390 y Fs(and)1479 2405 y Fq(S)1529 2390 y Ft(N)1607 2405 y Fq(T)1687 2390 y Fs(b)g(e)25 b(bimo)-5 b(dules.)41 b(Then)2524 2405 y Fq(R)2581 2390 y Fx(\()p Ft(M)11 b Fu(\012)2801 2405 y Fq(S)2853 2390 y Ft(N)f Fx(\))2979 2405 y Fq(T)3059 2390 y Fs(is)25 b(a)g(bimo)-5 b(dule)0 2506 y(by)35 b Ft(r)s Fx(\()p Ft(m)22 b Fu(\012)h Ft(n)p Fx(\))28 b(:=)f Ft(r)s(m)22 b Fu(\012)h Ft(n)35 b Fs(and)f Fx(\()p Ft(m)23 b Fu(\012)f Ft(n)p Fx(\))p Ft(t)28 b Fx(:=)g Ft(m)23 b Fu(\012)f Ft(nt)p Fs(.)149 2668 y Fi(Pr)n(oof.)42 b Fx(Ob)m(viously)i(w)m(e)h(ha)m(v)m(e)g(2.-4.)76 b(F)-8 b(urthermore)44 b(\()p Ft(r)32 b Fu(\012)2424 2683 y Fq(S)2505 2668 y Fx(id)o(\)\()p Ft(m)e Fu(\012)h Ft(n)p Fx(\))46 b(=)h Ft(r)s(m)30 b Fu(\012)g Ft(n)47 b Fx(=)0 2784 y Ft(r)s Fx(\()p Ft(m)22 b Fu(\012)h Ft(n)p Fx(\))32 b(is)g(a)h(homomorphism.)p 3529 2784 V 3533 2721 59 4 v 3533 2784 V 3591 2784 4 66 v 149 2945 a Fv(Corollary)j(8.4.13.)50 b Fs(Given)30 b(bimo)-5 b(dules)1736 2960 y Fq(R)1794 2945 y Ft(M)1888 2960 y Fq(S)1939 2945 y Fs(,)2001 2960 y Fq(S)2052 2945 y Ft(N)2130 2960 y Fq(T)2185 2945 y Fs(,)2246 2960 y Fq(R)2304 2945 y Ft(M)2408 2909 y Fr(0)2398 2971 y Fq(S)2449 2945 y Fs(,)2511 2960 y Fq(S)2562 2945 y Ft(N)2650 2909 y Fr(0)2640 2971 y Fq(T)2726 2945 y Fs(and)30 b(homomorphisms)0 3062 y Ft(f)40 b Fu(2)29 b Fx(Hom)386 3077 y Fq(R)p Fs(-)q Fq(S)526 3062 y Fx(\()p Ft(:M)5 b(:;)17 b(:M)892 3025 y Fr(0)916 3062 y Ft(:)p Fx(\))36 b Fs(and)f Ft(g)d Fu(2)e Fx(Hom)1585 3077 y Fq(S)t Fs(-)o Fq(T)1721 3062 y Fx(\()p Ft(:N)5 b(:;)17 b(:N)2055 3025 y Fr(0)2079 3062 y Ft(:)p Fx(\))p Fs(.)47 b(Then)35 b(we)g(have)g Ft(f)f Fu(\012)3005 3077 y Fq(S)3078 3062 y Ft(g)f Fu(2)c Fx(Hom)3457 3077 y Fq(R)p Fs(-)p Fq(T)0 3178 y Fx(\()p Ft(:M)k Fu(\012)269 3193 y Fq(S)342 3178 y Ft(N)5 b(:;)17 b(:M)627 3142 y Fr(0)674 3178 y Fu(\012)751 3193 y Fq(S)824 3178 y Ft(N)912 3142 y Fr(0)936 3178 y Ft(:)p Fx(\))p Ft(:)149 3339 y Fi(Pr)n(oof.)42 b Ft(f)33 b Fu(\012)692 3354 y Fq(S)766 3339 y Ft(g)t Fx(\()p Ft(r)s(m)21 b Fu(\012)i Ft(nt)p Fx(\))28 b(=)f Ft(f)11 b Fx(\()p Ft(r)s(m)p Fx(\))22 b Fu(\012)h Ft(g)t Fx(\()p Ft(nt)p Fx(\))k(=)g Ft(r)s Fx(\()p Ft(f)33 b Fu(\012)2352 3354 y Fq(S)2425 3339 y Ft(g)t Fx(\)\()p Ft(m)22 b Fu(\012)h Ft(n)p Fx(\))p Ft(t:)p 3529 3339 V 3533 3277 59 4 v 3533 3339 V 3591 3339 4 66 v 149 3501 a Fv(Remark)37 b(8.4.14.)50 b Fx(Ev)m(ery)33 b(mo)s(dule)e Ft(M)43 b Fx(o)m(v)m(er)33 b(a)e(comm)m(utativ)m(e)g(ring)g Fp(K)55 b Fx(and)33 b(in)e(particular)0 3617 y(ev)m(ery)42 b(v)m(ector)e(space)h(o)m(v)m(er)g(a)e(\014eld)h Fp(K)63 b Fx(is)39 b(a)h Fp(K)17 b Fx(-)p Fp(K)23 b Fx(-bimo)s(dule)43 b(b)m(y)e Ft(\025m)f Fx(=)g Ft(m\025)p Fx(.)65 b(So)39 b(there)i(is)e(an)0 3733 y(em)m(b)s(edding)28 b(functor)h Ft(\023)f Fx(:)g Fp(K)17 b Fx(-)6 b Fv(Mo)s(d)28 b Fu(\000)-60 b(!)28 b Fp(K)17 b Fx(-)6 b Fv(Mo)s(d)p Fx(-)p Fp(K)17 b Fx(.)48 b(Observ)m(e)31 b(that)d(there)h(are)g Fp(K)17 b Fx(-)p Fp(K)23 b Fx(-bimo)s(dules)0 3849 y(that)32 b(do)g(not)h(satisfy)f Ft(\025m)c Fx(=)f Ft(m\025)p Fx(.)44 b(T)-8 b(ak)m(e)34 b(for)e(example)f(an)i(automorphism)d Ft(\013)e Fx(:)g Fp(K)51 b Fu(\000)-60 b(!)28 b Fp(K)56 b Fx(and)32 b(a)0 3966 y(left)g Fp(K)17 b Fx(-mo)s(dule)37 b Ft(M)43 b Fx(and)33 b(de\014ne)h Ft(m\025)28 b Fx(:=)f Ft(\013)q Fx(\()p Ft(\025)p Fx(\))p Ft(m)p Fx(.)43 b(Then)34 b Ft(M)43 b Fx(is)33 b(suc)m(h)h(a)e Fp(K)17 b Fx(-)p Fp(K)23 b Fx(-bimo)s(dule.)149 4082 y(The)41 b(tensor)f(pro)s(duct)g Ft(M)e Fu(\012)1239 4097 y Fe(K)1322 4082 y Ft(N)50 b Fx(of)39 b(t)m(w)m(o)h Fp(K)18 b Fx(-)p Fp(K)23 b Fx(-bimo)s(dules)43 b Ft(M)51 b Fx(and)39 b Ft(N)50 b Fx(is)40 b(again)e(a)h Fp(K)17 b Fx(-)p Fp(K)23 b Fx(-)0 4198 y(bimo)s(dule.)41 b(If)31 b(w)m(e)h(ha)m(v)m(e,)h(ho)m(w)m(ev)m(er,)h Fp(K)18 b Fx(-)p Fp(K)23 b Fx(-bimo)s(dules)35 b Ft(M)42 b Fx(and)31 b Ft(N)42 b Fx(arising)29 b(from)h Fp(K)18 b Fx(-mo)s(dules)36 b(as)0 4314 y(ab)s(o)m(v)m(e,)d(i.e.)43 b(satisfying)30 b Ft(\025m)e Fx(=)g Ft(m\025)p Fx(,)k(then)g(their)g(tensor)g(pro)s (duct)g Ft(M)f Fu(\012)2692 4329 y Fe(K)2769 4314 y Ft(N)42 b Fx(also)31 b(satis\014es)h(this)0 4430 y(equation,)37 b(so)g Ft(M)f Fu(\012)761 4445 y Fe(K)842 4430 y Ft(N)47 b Fx(comes)37 b(from)e(a)h(mo)s(dule)g(in)f Fp(K)18 b Fx(-)p Fv(Mo)s(d)6 b Fx(.)55 b(Indeed)38 b(w)m(e)g(ha)m(v)m(e)g Ft(\025m)25 b Fu(\012)g Ft(n)35 b Fx(=)0 4547 y Ft(m\025)22 b Fu(\012)h Ft(n)28 b Fx(=)f Ft(m)c Fu(\012)g Ft(\025n)k Fx(=)h Ft(m)22 b Fu(\012)h Ft(n\025)p Fx(.)44 b(Th)m(us)34 b(the)f(follo)m(wing)d(diagram)g(of)i(functors)h(comm)m(utes:)991 5223 y Fp(K)18 b Fx(-)5 b Fv(Mo)s(d)764 b Fp(K)18 b Fx(-)5 b Fv(Mo)s(d)q Fx(-)p Fp(K)17 b Ft(:)p 1358 5190 706 4 v 1981 5188 a Fl(-)1694 5250 y Ft(\023)761 4731 y Fp(K)h Fx(-)5 b Fv(Mo)s(d)23 b Fu(\002)g Fp(K)17 b Fx(-)6 b Fv(Mo)s(d)263 b Fp(K)18 b Fx(-)5 b Fv(Mo)s(d)q Fx(-)p Fp(K)45 b Fu(\002)23 b Fp(K)17 b Fx(-)6 b Fv(Mo)s(d)q Fx(-)o Fp(K)p 1587 4703 206 4 v 1710 4701 a Fl(-)1594 4673 y Ft(\023)23 b Fu(\002)f Ft(\023)p 1158 5120 4 351 v 1160 5120 a Fl(?)1023 4959 y Fr(\012)1078 4970 y Fd(K)p 2328 5120 V 2330 5120 a Fl(?)2369 4959 y Fr(\012)2424 4970 y Fd(K)p eop %%Page: 14 14 14 13 bop 0 236 a Fo(14)1456 b(8.)77 b(TOOLBO)n(X)0 425 y Fx(So)45 b(w)m(e)g(can)g(consider)h Fp(K)17 b Fx(-)p Fv(Mo)s(d)51 b Fx(as)45 b(a)f(\(prop)s(er\))h(sub)s(category)h(of)e Fp(K)17 b Fx(-)6 b Fv(Mo)s(d)q Fx(-)o Fp(K)18 b Fx(.)86 b(The)45 b(tensor)0 541 y(pro)s(duct)33 b(o)m(v)m(er)g Fp(K)56 b Fx(can)33 b(b)s(e)g(restricted)g(to)f Fp(K)18 b Fx(-)p Fv(Mo)s(d)6 b Fx(.)149 658 y(W)-8 b(e)33 b(write)g(the)g (tensor)g(pro)s(duct)g(of)f(t)m(w)m(o)h(v)m(ector)h(spaces)g Ft(M)43 b Fx(and)33 b Ft(N)43 b Fx(as)33 b Ft(M)g Fu(\012)22 b Ft(N)10 b Fx(.)149 847 y Fv(Theorem)38 b(8.4.15.)49 b Fs(In)34 b(the)h(c)-5 b(ate)g(gory)35 b Fp(K)17 b Fs(-)p Fv(Mo)t(d)41 b Fs(ther)-5 b(e)35 b(ar)-5 b(e)34 b(natur)-5 b(al)35 b(isomorphisms)149 997 y Fx(1.)42 b(Asso)s(ciativit)m(y)31 b(La)m(w:)46 b Ft(\013)28 b Fx(:)g(\()p Ft(M)33 b Fu(\012)22 b Ft(N)10 b Fx(\))23 b Fu(\012)g Ft(P)1862 969 y Fu(\030)1863 1001 y Fx(=)1967 997 y Ft(M)33 b Fu(\012)22 b Fx(\()p Ft(N)33 b Fu(\012)23 b Ft(P)14 b Fx(\))p Fs(.)149 1113 y Fx(2.)42 b(La)m(w)33 b(of)f(the)h(Left)f(Unit:)45 b Ft(\025)27 b Fx(:)h Fp(K)46 b Fu(\012)22 b Ft(M)1696 1085 y Fu(\030)1697 1117 y Fx(=)1801 1113 y Ft(M)10 b Fs(.)149 1229 y Fx(3.)42 b(La)m(w)33 b(of)f(the)h(Righ)m(t)f(Unit:)44 b Ft(\032)28 b Fx(:)g Ft(M)k Fu(\012)23 b Fp(K)1754 1201 y Fu(\030)1755 1233 y Fx(=)1859 1229 y Ft(M)10 b Fs(.)149 1345 y Fx(4.)42 b(Symmetry)32 b(La)m(w:)46 b Ft(\034)39 b Fx(:)28 b Ft(M)33 b Fu(\012)22 b Ft(N)1466 1318 y Fu(\030)1467 1349 y Fx(=)1572 1345 y Ft(N)32 b Fu(\012)23 b Ft(M)10 b Fs(.)149 1462 y Fx(5.)42 b(Existence)34 b(of)e(Inner)h(Hom-F)-8 b(unctors:)44 b(Hom\()p Ft(P)35 b Fu(\012)23 b Ft(M)5 b(;)17 b(N)10 b Fx(\))2488 1434 y Fu(\030)2489 1466 y Fx(=)2593 1462 y(Hom)o(\()p Ft(P)s(;)17 b Fx(Hom)o(\()p Ft(M)5 b(;)17 b(N)10 b Fx(\)\))p Fs(.)149 1651 y Fi(Pr)n(oof.)42 b Fx(W)-8 b(e)33 b(only)f(describ)s(e)i(the)f(corresp)s(onding)f (homomorphisms.)149 1767 y(1.)44 b(Use)33 b(\(8.4.45.\))43 b(to)32 b(de\014ne)i Ft(\013)q Fx(\(\()p Ft(m)22 b Fu(\012)g Ft(n)p Fx(\))h Fu(\012)f Ft(p)p Fx(\))28 b(:=)g Ft(m)22 b Fu(\012)h Fx(\()p Ft(n)f Fu(\012)h Ft(p)p Fx(\).)149 1883 y(2.)44 b(De\014ne)33 b Ft(\025)27 b Fx(:)h Fp(K)46 b Fu(\012)22 b Ft(M)39 b Fu(\000)-60 b(!)28 b Ft(M)43 b Fx(b)m(y)33 b Ft(\025)p Fx(\()p Ft(r)25 b Fu(\012)e Ft(m)p Fx(\))28 b(:=)f Ft(r)s(m)p Fx(.)149 1999 y(3.)44 b(De\014ne)33 b Ft(\032)28 b Fx(:)g Ft(M)k Fu(\012)23 b Fp(K)51 b Fu(\000)-60 b(!)28 b Ft(M)43 b Fx(b)m(y)34 b Ft(\032)p Fx(\()p Ft(m)22 b Fu(\012)h Ft(r)s Fx(\))k(:=)h Ft(mr)s Fx(.)149 2116 y(4.)44 b(De\014ne)33 b Ft(\034)11 b Fx(\()p Ft(m)23 b Fu(\012)f Ft(n)p Fx(\))28 b(:=)g Ft(n)22 b Fu(\012)h Ft(m)p Fx(.)149 2232 y(5.)83 b(F)-8 b(or)45 b Ft(f)61 b Fx(:)50 b Ft(P)45 b Fu(\012)32 b Ft(M)60 b Fu(\000)-60 b(!)51 b Ft(N)56 b Fx(de\014ne)47 b Ft(\036)p Fx(\()p Ft(f)11 b Fx(\))49 b(:)i Ft(P)63 b Fu(\000)-60 b(!)50 b Fx(Hom\()p Ft(M)5 b(;)17 b(N)10 b Fx(\))46 b(b)m(y)h Ft(\036)p Fx(\()p Ft(f)11 b Fx(\)\()p Ft(p)p Fx(\)\()p Ft(m)p Fx(\))49 b(:=)0 2348 y Ft(f)11 b Fx(\()p Ft(p)22 b Fu(\012)g Ft(m)p Fx(\).)p 3529 2348 4 66 v 3533 2286 59 4 v 3533 2348 V 3591 2348 4 66 v 149 2546 a(Usually)36 b(one)h(iden)m(ti\014es)g(threefold)f(tensor)h (pro)s(ducts)g(along)e(the)i(map)f Ft(\013)h Fx(so)g(that)f(w)m(e)i (use)0 2662 y Ft(M)29 b Fu(\012)18 b Ft(N)29 b Fu(\012)18 b Ft(P)41 b Fx(=)28 b(\()p Ft(M)g Fu(\012)19 b Ft(N)10 b Fx(\))18 b Fu(\012)g Ft(P)42 b Fx(=)27 b Ft(M)i Fu(\012)18 b Fx(\()p Ft(N)29 b Fu(\012)18 b Ft(P)c Fx(\).)42 b(F)-8 b(or)30 b(the)h(notion)e(of)h(a)h(monoidal)c(or)j(tensor)0 2779 y(category)-8 b(,)33 b(ho)m(w)m(ev)m(er,)i(this)d(natural)g (transformation)e(is)i(of)g(cen)m(tral)h(imp)s(ortance.)149 2968 y Fv(Problem)j(8.4.4.)49 b Fx(1.)44 b(Giv)m(e)32 b(an)g(explicit)g(pro)s(of)f(of)h Ft(M)h Fu(\012)23 b Fx(\()p Ft(X)30 b Fu(\010)22 b Ft(Y)g Fx(\))2725 2940 y Fu(\030)2726 2972 y Fx(=)2830 2968 y Ft(M)33 b Fu(\012)23 b Ft(X)30 b Fu(\010)23 b Ft(M)32 b Fu(\012)23 b Ft(Y)e Fx(.)149 3084 y(2.)42 b(Sho)m(w)28 b(that)g(for)e(ev)m(ery)k(\014nite)d (dimensional)e(v)m(ector)j(space)h Ft(V)49 b Fx(there)28 b(is)f(a)g(unique)h(elemen)m(t)0 3126 y Ff(P)105 3152 y Fq(n)105 3230 y(i)p Fn(=1)240 3200 y Ft(v)287 3215 y Fq(i)338 3200 y Fu(\012)22 b Ft(v)488 3164 y Fr(\003)484 3225 y Fq(i)555 3200 y Fu(2)28 b Ft(V)44 b Fu(\012)22 b Ft(V)928 3164 y Fr(\003)1000 3200 y Fx(suc)m(h)34 b(that)e(the)h (follo)m(wing)d(holds)1226 3409 y Fu(8)p Ft(v)i Fu(2)c Ft(V)49 b Fx(:)1712 3315 y Ff(X)1772 3525 y Fq(i)1873 3409 y Ft(v)1924 3368 y Fr(\003)1920 3434 y Fq(i)1963 3409 y Fx(\()p Ft(v)t Fx(\))p Ft(v)2137 3424 y Fq(i)2193 3409 y Fx(=)27 b Ft(v)t(:)0 3684 y Fx(\(Hin)m(t:)44 b(Use)35 b(an)e(isomorphism)e(End\()p Ft(V)22 b Fx(\))1553 3657 y Fu(\030)1554 3689 y Fx(=)1659 3684 y Ft(V)44 b Fu(\012)24 b Ft(V)1939 3648 y Fr(\003)2012 3684 y Fx(and)33 b(dual)g(bases)h Fu(f)p Ft(v)2773 3699 y Fq(i)2801 3684 y Fu(g)f Fx(of)g Ft(V)55 b Fx(and)33 b Fu(f)p Ft(v)3399 3648 y Fr(\003)3395 3709 y Fq(i)3438 3684 y Fu(g)g Fx(of)0 3801 y Ft(V)79 3764 y Fr(\003)118 3801 y Fx(.\))149 3917 y(3.)41 b(Sho)m(w)25 b(that)f(the)h(follo)m(wing)d(diagrams)g(\(coherence)k(diagrams)d(or)h (constrain)m(ts\))h(comm)m(ute)0 4033 y(in)32 b Fp(K)17 b Fx(-)p Fv(Mo)s(d)7 b Fx(:)20 4219 y(\(\()p Ft(A)22 b Fu(\012)h Ft(B)5 b Fx(\))22 b Fu(\012)h Ft(C)7 b Fx(\))22 b Fu(\012)g Ft(D)538 b Fx(\()p Ft(A)22 b Fu(\012)h Fx(\()p Ft(B)k Fu(\012)c Ft(C)7 b Fx(\)\))22 b Fu(\012)g Ft(D)p 879 4196 478 4 v 1274 4194 a Fl(-)921 4158 y Fq(\013)p Fn(\()p Fq(A;B)s(;C)5 b Fn(\))p Fr(\012)p Fn(1)2750 4219 y Ft(A)22 b Fu(\012)h Fx(\(\()p Ft(B)k Fu(\012)c Ft(C)7 b Fx(\))22 b Fu(\012)g Ft(D)s Fx(\))p 2244 4196 V 2639 4194 a Fl(-)2275 4158 y Fq(\013)p Fn(\()p Fq(A;B)s Fr(\012)p Fq(C)q(;D)r Fn(\))p 433 4516 4 254 v 435 4516 a Fl(?)474 4406 y Fq(\013)p Fn(\()p Fq(A)p Fr(\012)p Fq(B)s(;C)q(;D)r Fn(\))p 3163 4516 V 3165 4516 a Fl(?)2729 4406 y Fn(1)p Fr(\012)p Fq(\013)p Fn(\()p Fq(B)s(;C)q(;D)r Fn(\))20 4609 y Fx(\()p Ft(A)g Fu(\012)h Ft(B)5 b Fx(\))22 b Fu(\012)h Fx(\()p Ft(C)29 b Fu(\012)22 b Ft(D)s Fx(\))1900 b Ft(A)22 b Fu(\012)h Fx(\()p Ft(B)k Fu(\012)c Fx(\()p Ft(C)29 b Fu(\012)22 b Ft(D)s Fx(\)\))p 879 4586 1843 4 v 2639 4584 a Fl(-)1591 4548 y Fq(\013)p Fn(\()p Fq(A;B)s(;C)5 b Fr(\012)p Fq(D)r Fn(\))940 4846 y Fx(\()p Ft(A)23 b Fu(\012)f Fp(K)c Fx(\))28 b Fu(\012)22 b Ft(B)626 b(A)23 b Fu(\012)f Fx(\()p Fp(K)46 b Fu(\012)22 b Ft(B)5 b Fx(\))p 1518 4822 564 4 v 1999 4820 a Fl(-)1650 4785 y Fq(\013)p Fn(\()p Fq(A;)p Fe(K)g Fq(;B)s Fn(\))1663 5241 y Ft(A)22 b Fu(\012)h Ft(B)1212 5032 y Fq(\032)p Fn(\()p Fq(A)p Fn(\))p Fr(\012)p Fn(1)1317 4944 y Fl(Q)1400 5000 y(Q)1483 5055 y(Q)1566 5110 y(Q)1615 5142 y(Q)-83 b(s)2154 5032 y Fn(1)p Fr(\012)p Fq(\025)p Fn(\()p Fq(B)s Fn(\))2200 4944 y Fl(\021)2117 5000 y(\021)2034 5055 y(\021)1951 5110 y(\021)1902 5142 y(\021)g(+)p eop %%Page: 15 15 15 14 bop 1348 236 a Fo(4.)77 b(TENSOR)32 b(PR)n(ODUCTS)1245 b(15)149 425 y Fx(4.)43 b(W)-8 b(rite)32 b Ft(\034)11 b Fx(\()p Ft(A;)17 b(B)5 b Fx(\))28 b(:)g Ft(A)21 b Fu(\012)g Ft(B)33 b Fu(\000)-60 b(!)28 b Ft(B)e Fu(\012)c Ft(A)32 b Fx(for)f Ft(\034)11 b Fx(\()p Ft(A;)17 b(B)5 b Fx(\))28 b(:)g Ft(a)21 b Fu(\012)h Ft(b)28 b Fu(7!)f Ft(b)22 b Fu(\012)f Ft(a)p Fx(.)44 b(Sho)m(w)32 b(that)g Ft(\034)44 b Fx(is)31 b(a)0 541 y(natural)g(transformation)g(\(b)s(et)m(w)m(een)j (whic)m(h)g(functors?\).)44 b(Sho)m(w)33 b(that)513 726 y(\()p Ft(A)22 b Fu(\012)h Ft(B)5 b Fx(\))22 b Fu(\012)g Ft(C)434 b Fx(\()p Ft(B)27 b Fu(\012)c Ft(A)p Fx(\))f Fu(\012)g Ft(C)p 1090 703 369 4 v 1376 701 a Fl(-)1115 665 y Fq(\034)8 b Fn(\()p Fq(X)q(;B)s Fn(\))p Fr(\012)p Fn(1)2463 726 y Ft(B)27 b Fu(\012)c Fx(\()p Ft(A)f Fu(\012)g Ft(C)7 b Fx(\))p 2065 703 V 2351 701 a Fl(-)2227 682 y Fq(\013)p 785 1120 4 351 v 787 1120 a Fl(?)703 959 y Fq(\013)p 2735 1120 V 2737 1120 a Fl(?)2776 961 y Fn(1)p Fr(\012)p Fq(\034)h Fn(\()p Fq(A;C)d Fn(\))513 1213 y Ft(A)22 b Fu(\012)h Fx(\()p Ft(B)k Fu(\012)22 b Ft(C)7 b Fx(\))427 b(\()p Ft(B)27 b Fu(\012)c Ft(C)7 b Fx(\))22 b Fu(\012)g Ft(A)p 1090 1191 369 4 v 1376 1189 a Fl(-)1108 1153 y Fq(\034)8 b Fn(\()p Fq(A;B)s Fr(\012)p Fq(C)d Fn(\))2463 1213 y Ft(B)27 b Fu(\012)c Fx(\()p Ft(C)29 b Fu(\012)22 b Ft(A)p Fx(\))p 2065 1191 V 2351 1189 a Fl(-)2227 1170 y Fq(\013)0 1356 y Fx(comm)m(utes)32 b(for)h(all)d Ft(A;)17 b(B)5 b(;)17 b(C)34 b Fu(2)28 b Fp(K)18 b Fx(-)5 b Fv(Mo)s(d)34 b Fx(and)e(that)1284 1513 y Ft(\034)11 b Fx(\()p Ft(B)5 b(;)17 b(A)p Fx(\))p Ft(\034)11 b Fx(\()p Ft(A;)17 b(B)5 b Fx(\))29 b(=)e(id)2147 1528 y Fq(A)p Fr(\012)p Fq(B)0 1670 y Fx(for)32 b(all)f Ft(A)p Fx(,)h Ft(B)38 b Fx(in)32 b Fp(K)17 b Fx(-)6 b Fv(Mo)s(d)p Fx(.)149 1787 y(5.)44 b(Find)31 b(an)i(example)f(of)g Ft(M)10 b Fx(,)33 b Ft(N)39 b Fu(2)28 b Fp(K)17 b Fx(-)6 b Fv(Mo)s(d)p Fx(-)p Fp(K)56 b Fx(suc)m(h)34 b(that)e Ft(M)h Fu(\012)2619 1802 y Fe(K)2698 1787 y Ft(N)38 b Fu(6)2814 1759 y(\030)2815 1791 y Fx(=)2919 1787 y Ft(N)33 b Fu(\012)3107 1802 y Fe(K)3185 1787 y Ft(M)10 b Fx(.)p eop %%Page: 16 16 16 15 bop 0 236 a Fo(16)1456 b(8.)77 b(TOOLBO)n(X)1495 425 y Fv(5.)93 b(Algebras)149 600 y Fx(Let)26 b Fp(K)50 b Fx(b)s(e)26 b(a)f(comm)m(utativ)m(e)g(ring.)40 b(In)26 b(most)f(of)h(our)f(applications)f Fp(K)49 b Fx(will)23 b(b)s(e)j(a)g(\014eld.)41 b(T)-8 b(ensor)0 716 y(pro)s(ducts)39 b(of)f Fp(K)18 b Fx(-mo)s(dules)43 b(will)36 b(b)s(e)i(simply)f (written)i(as)f Ft(M)f Fu(\012)27 b Ft(N)48 b Fx(:=)38 b Ft(M)f Fu(\012)2855 731 y Fq(K)2950 716 y Ft(N)10 b Fx(.)61 b(Ev)m(ery)40 b(suc)m(h)0 832 y(tensor)25 b(pro)s(duct)f(is)f (again)g(a)g Fp(K)18 b Fx(-bimo)s(dule)27 b(since)e(eac)m(h)f Fp(K)18 b Fx(-mo)s(dule)28 b Ft(M)35 b Fx(resp.)25 b Ft(N)34 b Fx(is)23 b(a)h Fp(K)18 b Fx(-bimo)s(dule)0 948 y(\(see)34 b(8.4.14\).)149 1128 y Fv(De\014nition)j(8.5.1.)49 b Fx(A)29 b Fp(K)17 b Fs(-algebr)-5 b(a)34 b Fx(is)29 b(a)g(v)m(ector)h(space)g Ft(A)f Fx(together)g(with)g(a)g Fs(multiplic)-5 b(ation)0 1244 y Fu(r)28 b Fx(:)f Ft(A)c Fu(\012)f Ft(A)28 b Fu(\000)-60 b(!)28 b Ft(A)33 b Fx(that)f(is)g(asso) s(ciativ)m(e:)1325 1901 y Ft(A)22 b Fu(\012)h Ft(A)609 b(A)p 1622 1873 552 4 v 2091 1871 a Fl(-)1868 1935 y Fr(r)1227 1413 y Ft(A)23 b Fu(\012)f Ft(A)g Fu(\012)h Ft(A)415 b(A)22 b Fu(\012)h Ft(A)p 1719 1385 357 4 v 1993 1383 a Fl(-)1805 1357 y Fn(id)12 b Fr(\012r)p 1457 1803 4 351 v 1459 1803 a Fl(?)1247 1647 y Fr(r\012)p Fn(id)p 2237 1803 V 2239 1803 a Fl(?)2278 1650 y Fr(r)0 2063 y Fx(and)33 b(a)f Fs(unit)h Ft(\021)e Fx(:)d Fp(K)51 b Fu(\000)-60 b(!)28 b Ft(A)p Fx(:)1025 2249 y Fp(K)45 b Fu(\012)23 b Ft(A)1325 2221 y Fu(\030)1326 2253 y Fx(=)1430 2249 y Ft(A)1531 2221 y Fu(\030)1532 2253 y Fx(=)1636 2249 y Ft(A)f Fu(\012)h Fp(K)423 b Ft(A)22 b Fu(\012)g Ft(A)p 1937 2221 342 4 v 2196 2219 a Fl(-)2026 2187 y Fn(id)12 b Fr(\012)p Fq(\021)p 1465 2638 4 351 v 1467 2638 a Fl(?)1276 2479 y Fq(\021)r Fr(\012)p Fn(id)p 2440 2638 V 2442 2638 a Fl(?)2481 2485 y Fr(r)1333 2736 y Ft(A)22 b Fu(\012)g Ft(A)791 b(A:)p 1629 2708 734 4 v 2280 2706 a Fl(-)1967 2771 y Fr(r)2016 2486 y Fn(id)1603 2329 y Fl(H)1686 2370 y(H)1769 2412 y(H)1852 2453 y(H)1935 2495 y(H)2018 2536 y(H)2101 2578 y(H)2184 2619 y(H)2222 2638 y(H)-83 b(j)0 2899 y Fx(A)33 b Fp(K)17 b Fx(-algebra)37 b Ft(A)c Fx(is)f Fs(c)-5 b(ommutative)32 b Fx(if)f(the)i(follo)m(wing)d (diagram)g(comm)m(utes)1422 3053 y Ft(A)23 b Fu(\012)f Ft(A)220 b(A)22 b Fu(\012)h Ft(A)p 1719 3025 162 4 v 1798 3023 a Fl(-)1781 3005 y Fq(\034)1750 3545 y Ft(A:)1560 3290 y Fr(r)1590 3175 y Fl(A)1632 3258 y(A)1673 3341 y(A)1715 3424 y(A)1724 3443 y(A)-42 b(U)1981 3290 y Fr(r)1968 3175 y Fl(\001)1927 3258 y(\001)1885 3341 y(\001)1844 3424 y(\001)1834 3443 y(\001)g(\013)0 3677 y Fx(Let)36 b Ft(A)f Fx(and)g Ft(B)41 b Fx(b)s(e)35 b Fp(K)18 b Fx(-algebras.)57 b(A)35 b Fs(homomorphism)g(of)i(algebr)-5 b(as)35 b Ft(f)43 b Fx(:)32 b Ft(A)h Fu(\000)-60 b(!)32 b Ft(B)41 b Fx(is)34 b(a)i Fp(K)17 b Fx(-linear)0 3794 y(map)32 b(suc)m(h)i(that)e(the)h (follo)m(wing)d(diagrams)h(comm)m(ute:)1469 4471 y Ft(A)509 b(B)p 1571 4439 452 4 v 1940 4437 a Fl(-)1776 4502 y Fq(f)1372 3980 y Ft(A)22 b Fu(\012)h Ft(A)311 b(B)27 b Fu(\012)c Ft(B)p 1669 3951 254 4 v 1840 3949 a Fl(-)1727 3917 y Fq(f)7 b Fr(\012)p Fq(f)p 1504 4369 4 351 v 1506 4369 a Fl(?)1358 4210 y Fr(r)1417 4221 y Fk(A)p 2089 4369 V 2091 4369 a Fl(?)2130 4210 y Fr(r)2189 4221 y Fk(B)0 4643 y Fx(and)1753 4729 y Fp(K)1525 4947 y Fq(\021)1560 4958 y Fk(A)1716 4846 y Fl(\001)1675 4929 y(\001)1633 5012 y(\001)1592 5095 y(\001)1582 5114 y(\001)-42 b(\013)1973 4947 y Fq(\021)2008 4958 y Fk(B)1826 4846 y Fl(A)1867 4929 y(A)1909 5012 y(A)1950 5095 y(A)1960 5114 y(A)g(U)1511 5216 y Ft(A)398 b(B)5 b(:)p 1613 5184 341 4 v 1871 5182 a Fl(-)1762 5150 y Fq(f)p eop %%Page: 17 17 17 16 bop 1530 236 a Fo(5.)77 b(ALGEBRAS)1427 b(17)149 425 y Fv(Remark)37 b(8.5.2.)50 b Fx(Ev)m(ery)34 b Fp(K)18 b Fx(-algebra)37 b Ft(A)32 b Fx(is)g(a)h(ring)e(with)h(the)h(m)m (ultiplication)1309 669 y Ft(A)23 b Fu(\002)f Ft(A)1636 591 y Fr(\012)1605 669 y Fu(\000)-60 b(!)28 b Ft(A)22 b Fu(\012)h Ft(A)2075 591 y Fr(r)2046 669 y Fu(\000)-60 b(!)27 b Ft(A:)0 868 y Fx(The)34 b(unit)d(elemen)m(t)i(is)f Ft(\021)t Fx(\(1\),)g(where)i(1)e(is)g(the)h(unit)f(elemen)m(t)g(of)h Fp(K)17 b Fx(.)149 984 y(Ob)m(viously)38 b(the)h(comp)s(osition)d(of)h (t)m(w)m(o)i(homomorphisms)d(of)h(algebras)h(is)f(again)g(a)g(homo-)0 1100 y(morphism)j(of)i(algebras.)72 b(F)-8 b(urthermore)42 b(the)g(iden)m(tit)m(y)g(map)g(is)f(a)h(homomorphism)e(of)h(alge-)0 1217 y(bras.)62 b(Hence)40 b(the)f Fp(K)18 b Fx(-algebras)43 b(form)37 b(a)i(category)g Fp(K)17 b Fx(-)p Fv(Alg)5 b Fx(.)61 b(The)40 b(category)f(of)f(comm)m(utativ)m(e)0 1333 y Fp(K)17 b Fx(-algebras)38 b(will)30 b(b)s(e)j(denoted)g(b)m(y)h Fp(K)17 b Fx(-)p Fv(cAlg)5 b Fx(.)149 1531 y Fv(Problem)36 b(8.5.5.)49 b Fx(1.)44 b(Sho)m(w)33 b(that)f(End)1676 1546 y Fq(K)1745 1531 y Fx(\()p Ft(V)21 b Fx(\))32 b(is)h(a)f Fp(K)17 b Fx(-algebra.)149 1647 y(2.)74 b(Sho)m(w)44 b(that)f(\()p Ft(A;)17 b Fu(r)45 b Fx(:)g Ft(A)29 b Fu(\012)h Ft(A)45 b Fu(\000)-60 b(!)45 b Ft(A;)17 b(\021)49 b Fx(:)c Fp(K)69 b Fu(\000)-60 b(!)45 b Ft(A)p Fx(\))e(is)f(a)g Fp(K)18 b Fx(-algebra)47 b(if)42 b(and)h(only)f(if)0 1814 y Ft(A)f Fx(with)g(the)g(m)m(ultiplication)c Ft(A)28 b Fu(\002)g Ft(A)1508 1736 y Fr(\012)1477 1814 y Fu(\000)-60 b(!)42 b Ft(A)28 b Fu(\012)h Ft(A)1987 1736 y Fr(r)1958 1814 y Fu(\000)-60 b(!)42 b Ft(A)f Fx(and)g(the)h(unit)e Ft(\021)t Fx(\(1\))h(is)f(a)h(ring)f(and)0 1930 y Ft(\021)31 b Fx(:)d Fp(K)51 b Fu(\000)-60 b(!)28 b Fx(Cen)m(t)q(\()p Ft(A)p Fx(\))33 b(is)f(a)g(ring)g(homomorphism)e(in)m(to)h(the)i(cen)m (ter)h(of)e Ft(A)p Fx(.)149 2047 y(3.)71 b(Let)42 b Ft(V)63 b Fx(b)s(e)42 b(a)f Fp(K)18 b Fx(-mo)s(dule.)75 b(Sho)m(w)42 b(that)g Ft(D)s Fx(\()p Ft(V)21 b Fx(\))43 b(:=)g Fp(K)52 b Fu(\002)29 b Ft(V)63 b Fx(with)41 b(the)i(m)m(ultiplication)0 2163 y(\()p Ft(r)82 2178 y Fn(1)121 2163 y Ft(;)17 b(v)212 2178 y Fn(1)252 2163 y Fx(\)\()p Ft(r)372 2178 y Fn(2)411 2163 y Ft(;)g(v)502 2178 y Fn(2)541 2163 y Fx(\))28 b(:=)f(\()p Ft(r)819 2178 y Fn(1)859 2163 y Ft(r)903 2178 y Fn(2)942 2163 y Ft(;)17 b(r)1030 2178 y Fn(1)1069 2163 y Ft(v)1116 2178 y Fn(2)1178 2163 y Fx(+)22 b Ft(r)1320 2178 y Fn(2)1359 2163 y Ft(v)1406 2178 y Fn(1)1446 2163 y Fx(\))32 b(is)h(a)f(comm)m (utativ)m(e)f Fp(K)18 b Fx(-algebra.)149 2361 y Fv(Lemma)38 b(8.5.3.)49 b Fs(L)-5 b(et)45 b Ft(A)h Fs(and)e Ft(B)50 b Fs(b)-5 b(e)45 b(algebr)-5 b(as.)75 b(Then)44 b Ft(A)30 b Fu(\012)g Ft(B)50 b Fs(is)45 b(an)g(algebr)-5 b(a)44 b(with)h(the)0 2477 y(multiplic)-5 b(ation)34 b Fx(\()p Ft(a)704 2492 y Fn(1)766 2477 y Fu(\012)23 b Ft(b)907 2492 y Fn(1)946 2477 y Fx(\)\()p Ft(a)1073 2492 y Fn(2)1135 2477 y Fu(\012)g Ft(b)1276 2492 y Fn(2)1316 2477 y Fx(\))k(:=)h Ft(a)1563 2492 y Fn(1)1603 2477 y Ft(a)1654 2492 y Fn(2)1715 2477 y Fu(\012)23 b Ft(b)1856 2492 y Fn(1)1896 2477 y Ft(b)1937 2492 y Fn(2)1977 2477 y Fs(.)149 2676 y Fi(Pr)n(oof.)42 b Fx(Certainly)37 b(the)h(algebra)f(prop)s(erties)h(can)g(easily)e(b)s (e)i(c)m(hec)m(k)m(ed)j(b)m(y)e(a)e(simple)f(cal-)0 2792 y(culation)31 b(with)h(elemen)m(ts.)44 b(W)-8 b(e)33 b(prefer)g(for)f(later)g(applications)e(a)i(diagrammatic)d(pro)s(of.) 149 2908 y(Let)39 b Fu(r)413 2923 y Fq(A)507 2908 y Fx(:)e Ft(A)26 b Fu(\012)g Ft(A)37 b Fu(\000)-60 b(!)38 b Ft(A)g Fx(and)g Fu(r)1427 2923 y Fq(B)1525 2908 y Fx(:)f Ft(B)31 b Fu(\012)26 b Ft(B)42 b Fu(\000)-60 b(!)37 b Ft(B)44 b Fx(denote)38 b(the)h(m)m(ultiplications)34 b(of)k(the)0 3024 y(t)m(w)m(o)i(algebras.)62 b(Then)40 b(the)f(new)h(m)m (ultiplication)35 b(is)j Fu(r)2090 3039 y Fq(A)p Fr(\012)p Fq(B)2297 3024 y Fx(:=)h(\()p Fu(r)2560 3039 y Fq(A)2643 3024 y Fu(\012)27 b(r)2830 3039 y Fq(B)2891 3024 y Fx(\)\(1)3016 3039 y Fq(A)3099 3024 y Fu(\012)g Ft(\034)38 b Fu(\012)27 b Fx(1)3436 3039 y Fq(B)3496 3024 y Fx(\))39 b(:)0 3141 y Ft(A)26 b Fu(\012)h Ft(B)k Fu(\012)26 b Ft(A)g Fu(\012)h Ft(B)42 b Fu(\000)-60 b(!)38 b Ft(A)26 b Fu(\012)g Ft(B)44 b Fx(where)39 b Ft(\034)49 b Fx(:)37 b Ft(B)32 b Fu(\012)26 b Ft(A)38 b Fu(\000)-60 b(!)37 b Ft(A)26 b Fu(\012)h Ft(B)43 b Fx(is)38 b(the)g(symmetry)h(map)e(from)0 3257 y(Theorem)c(8.4.15.)43 b(No)m(w)33 b(the)g(follo)m(wing)d(diagrams)h (comm)m(ute)1 3479 y Ft(A)22 b Fu(\012)h Ft(B)k Fu(\012)c Ft(A)f Fu(\012)h Ft(B)k Fu(\012)c Ft(A)f Fu(\012)h Ft(B)402 b(A)22 b Fu(\012)h Ft(A)f Fu(\012)g Ft(B)28 b Fu(\012)22 b Ft(B)28 b Fu(\012)22 b Ft(A)g Fu(\012)h Ft(B)p 1096 3451 340 4 v 1353 3449 a Fl(-)1138 3423 y Fn(1)p Fr(\012)p Fq(\034)8 b Fr(\012)p Fn(1)1357 3399 y Fc(3)2929 3479 y Ft(A)22 b Fu(\012)h Ft(B)k Fu(\012)c Ft(A)f Fu(\012)h Ft(B)p 2558 3451 343 4 v 2818 3449 a Fl(-)2580 3423 y Fr(r\012r\012)p Fn(1)2843 3399 y Fc(2)1 4210 y Ft(A)f Fu(\012)h Ft(B)k Fu(\012)c Ft(A)f Fu(\012)h Ft(A)f Fu(\012)h Ft(B)k Fu(\012)c Ft(B)402 b(A)22 b Fu(\012)h Ft(A)f Fu(\012)g Ft(A)h Fu(\012)f Ft(B)28 b Fu(\012)22 b Ft(B)27 b Fu(\012)c Ft(B)p 1096 4182 340 4 v 1353 4180 a Fl(-)1036 4255 y Fn(1)p Fr(\012)p Fq(\034)1157 4266 y Fk(B)r(;A)p Fh(\012)p Fk(A)1369 4255 y Fr(\012)p Fn(1)1459 4232 y Fc(3)2929 4210 y Ft(A)f Fu(\012)h Ft(A)f Fu(\012)h Ft(B)k Fu(\012)c Ft(B)p 2558 4182 343 4 v 2818 4180 a Fl(-)2552 4155 y Fr(r\012)p Fn(1)p Fr(\012r\012)p Fn(1)p 532 4112 4 595 v 534 4112 a Fl(?)241 3839 y Fn(1)276 3816 y Fc(3)311 3839 y Fr(\012)p Fq(\034)8 b Fr(\012)p Fn(1)p 1995 4112 V 1996 4112 a Fl(?)2035 3826 y Fn(1)p Fr(\012)p Fq(\034)2156 3837 y Fk(B)r Fh(\012)p Fk(B)r(;A)2371 3826 y Fr(\012)p Fn(1)p 3165 4112 V 3166 4112 a Fl(?)3205 3833 y Fn(1)p Fr(\012)p Fq(\034)g Fr(\012)p Fn(1)732 3845 y Ft(A)23 b Fu(\012)f Ft(A)h Fu(\012)f Ft(B)27 b Fu(\012)c Ft(A)f Fu(\012)h Ft(B)k Fu(\012)c Ft(B)583 4021 y Fn(1)p Fr(\012)p Fq(\034)8 b Fr(\012)p Fn(1)802 3998 y Fc(3)671 4112 y Fl(\010)754 4071 y(\010)837 4029 y(\010)920 3988 y(\010)1003 3946 y(\010)1046 3925 y(\010)-83 b(*)1315 3656 y Fn(1)1350 3632 y Fc(3)1384 3656 y Fr(\012)p Fq(\034)8 b Fr(\012)p Fn(1)1777 3559 y Fl(\010)1694 3600 y(\010)1611 3642 y(\010)1528 3684 y(\010)1445 3725 y(\010)1402 3747 y(\010)-83 b(\031)1693 4021 y Fn(1)1728 3998 y Fc(2)1763 4021 y Fr(\012)p Fq(\034)8 b Fr(\012)p Fn(1)1947 3998 y Fc(2)1402 3925 y Fl(H)1485 3966 y(H)1568 4008 y(H)1651 4049 y(H)1734 4091 y(H)1777 4112 y(H)-83 b(j)199 4698 y Ft(A)22 b Fu(\012)h Ft(B)k Fu(\012)c Ft(A)f Fu(\012)h Ft(B)798 b(A)22 b Fu(\012)g Ft(A)h Fu(\012)f Ft(B)28 b Fu(\012)22 b Ft(B)p 898 4670 736 4 v 1551 4668 a Fl(-)1156 4642 y Fn(1)p Fr(\012)p Fq(\034)8 b Fr(\012)p Fn(1)3127 4698 y Ft(A)22 b Fu(\012)h Ft(B)p 2360 4670 739 4 v 3016 4668 a Fl(-)2642 4642 y Fr(r\012r)p 532 4600 4 351 v 534 4600 a Fl(?)232 4443 y Fn(1)p Fr(\012r\012r)p 1995 4600 V 1996 4600 a Fl(?)2035 4443 y Fn(1)p Fr(\012r\012)p Fn(1)p Fr(\012r)p 3165 4600 V 3166 4600 a Fl(?)3205 4443 y Fr(r\012r)0 4867 y Fx(In)k(the)h(left)e (upp)s(er)h(rectangle)g(of)g(the)g(diagram)e(the)i(quadrangle)g(comm)m (utes)g(b)m(y)h(the)f(prop)s(erties)0 4983 y(of)36 b(the)g(tensor)h (pro)s(duct)f(and)h(the)f(t)m(w)m(o)h(triangles)e(comm)m(ute)h(b)m(y)h (inner)f(prop)s(erties)g(of)f Ft(\034)11 b Fx(.)55 b(The)0 5099 y(righ)m(t)26 b(upp)s(er)i(and)f(left)f(lo)m(w)m(er)h(rectangles)g (comm)m(ute)f(since)i Ft(\034)38 b Fx(is)27 b(a)f(natural)g (transformation)f(and)0 5216 y(the)33 b(righ)m(t)f(lo)m(w)m(er)g (rectangle)h(comm)m(utes)f(b)m(y)i(the)f(asso)s(ciativit)m(y)e(of)h (the)h(algebras)f Ft(A)h Fx(and)f Ft(B)5 b Fx(.)p eop %%Page: 18 18 18 17 bop 0 236 a Fo(18)1456 b(8.)77 b(TOOLBO)n(X)149 425 y Fx(F)-8 b(urthermore)30 b(w)m(e)i(use)f(the)g(homomorphism)d Ft(\021)j Fx(=)c Ft(\021)2103 440 y Fq(A)p Fr(\012)p Fq(B)2300 425 y Fx(:)g Fp(K)52 b Fu(\000)-60 b(!)27 b Fp(K)42 b Fu(\012)18 b Ft(K)35 b Fu(\000)-60 b(!)27 b Ft(A)18 b Fu(\012)g Ft(B)36 b Fx(in)30 b(the)0 541 y(follo)m(wing)g (comm)m(utativ)m(e)h(diagram)82 752 y Fp(K)46 b Fu(\012)23 b Ft(A)f Fu(\012)h Ft(B)583 724 y Fu(\030)584 756 y Fx(=)689 752 y Ft(A)f Fu(\012)g Ft(B)990 724 y Fu(\030)991 756 y Fx(=)1095 752 y Ft(A)h Fu(\012)f Ft(B)28 b Fu(\012)22 b Fp(K)407 b Ft(A)22 b Fu(\012)h Ft(B)k Fu(\012)22 b Fp(K)46 b Fu(\012)23 b Fp(K)p 1598 723 326 4 v 1841 721 a Fl(-)2928 752 y Ft(A)f Fu(\012)h Ft(B)k Fu(\012)c Ft(A)f Fu(\012)h Ft(B)p 2653 723 247 4 v 2817 721 a Fl(-)2648 689 y Fn(1)2683 666 y Fc(2)2718 689 y Fr(\012)p Fq(\021)r Fr(\012)p Fq(\021)2107 1352 y Fn(1)962 831 y Fl(H)1045 873 y(H)1128 914 y(H)1211 956 y(H)1294 997 y(H)1377 1039 y(H)1460 1080 y(H)1543 1122 y(H)1626 1163 y(H)1709 1205 y(H)1792 1246 y(H)1875 1288 y(H)1958 1330 y(H)2041 1371 y(H)2124 1413 y(H)2207 1454 y(H)2290 1496 y(H)2373 1537 y(H)2456 1579 y(H)2539 1620 y(H)2622 1662 y(H)2706 1703 y(H)2789 1745 y(H)2872 1786 y(H)2955 1828 y(H)3038 1869 y(H)3043 1872 y(H)-83 b(j)1450 831 y(H)1533 873 y(H)1616 914 y(H)1699 956 y(H)1782 997 y(H)1865 1039 y(H)1948 1080 y(H)2031 1122 y(H)2068 1141 y(H)g(j)1952 1239 y Ft(A)22 b Fu(\012)h Fp(K)45 b Fu(\012)23 b Ft(B)k Fu(\012)c Fp(K)327 b Ft(A)22 b Fu(\012)h Ft(A)f Fu(\012)h Ft(B)k Fu(\012)c Ft(B)p 2653 1211 V 2817 1209 a Fl(-)2620 1271 y Fn(1)p Fr(\012)p Fq(\021)r Fr(\012)p Fn(1)p Fr(\012)p Fq(\021)p 2286 1141 4 351 v 2288 1141 a Fl(?)2327 983 y Fn(1)p Fr(\012)p Fq(\034)8 b Fr(\012)p Fn(1)p 3261 1141 V 3263 1141 a Fl(?)3302 983 y Fn(1)p Fr(\012)p Fq(\034)g Fr(\012)p Fn(1)p 3261 1872 4 595 v 3263 1872 a Fl(?)3302 1594 y Fr(r\012r)475 831 y Fl(H)558 873 y(H)641 914 y(H)724 956 y(H)807 997 y(H)890 1039 y(H)973 1080 y(H)1056 1122 y(H)1094 1141 y(H)-83 b(j)2 1239 y Fp(K)45 b Fu(\012)23 b Fp(K)46 b Fu(\012)22 b Ft(A)h Fu(\012)f Ft(B)308 b Fp(K)45 b Fu(\012)23 b Ft(A)f Fu(\012)h Fp(K)46 b Fu(\012)22 b Ft(B)p 703 1211 246 4 v 866 1209 a Fl(-)716 1183 y Fn(1)p Fr(\012)p Fq(\034)8 b Fr(\012)p Fn(1)p 336 1141 4 351 v 338 1141 a Fl(?)3 1970 y Ft(A)22 b Fu(\012)h Ft(B)k Fu(\012)c Ft(A)f Fu(\012)h Ft(B)310 b(A)22 b Fu(\012)h Ft(A)f Fu(\012)h Ft(B)k Fu(\012)c Ft(B)p 702 1942 248 4 v 867 1940 a Fl(-)716 1914 y Fn(1)p Fr(\012)p Fq(\034)8 b Fr(\012)p Fn(1)3112 1970 y Ft(A)23 b Fu(\012)f Ft(B)5 b(:)p 1677 1942 1407 4 v 3001 1940 a Fl(-)2293 1915 y Fr(r\012r)p 336 1872 4 595 v 338 1872 a Fl(?)44 1596 y Fq(\021)r Fr(\012)p Fq(\021)r Fr(\012)p Fn(1)263 1573 y Fc(2)p 1311 1872 V 1313 1872 a Fl(?)963 1590 y Fq(\021)r Fr(\012)p Fn(1)p Fr(\012)p Fq(\021)r Fr(\012)p Fn(1)p 3529 2121 4 66 v 3533 2059 59 4 v 3533 2121 V 3591 2121 4 66 v 149 2309 a Fv(De\014nition)37 b(8.5.4.)49 b Fx(Let)38 b Fp(K)62 b Fx(b)s(e)38 b(a)g(comm)m(utativ)m(e)f(ring.)60 b(Let)38 b Ft(V)60 b Fx(b)s(e)38 b(a)g Fp(K)17 b Fx(-mo)s(dule.)65 b(A)38 b Fp(K)17 b Fx(-)0 2426 y(algebra)35 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))35 b(together)h(with)g(a)f(homomorphism)e(of) i Fp(K)18 b Fx(-mo)s(dules)40 b Ft(\023)33 b Fx(:)h Ft(V)54 b Fu(\000)-60 b(!)33 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))36 b(is)f(called)0 2542 y(a)j Fs(tensor)i(algebr)-5 b(a)39 b(over)f Ft(V)59 b Fx(if)37 b(for)h(eac)m(h)h Fp(K)18 b Fx(-algebra)42 b Ft(A)d Fx(and)f(for)g(eac)m(h)h(homomorphism)c(of)j Fp(K)17 b Fx(-)0 2658 y(mo)s(dules)22 b Ft(f)38 b Fx(:)28 b Ft(V)49 b Fu(\000)-60 b(!)28 b Ft(A)22 b Fx(there)h(exists)h(a)e (unique)h(homomorphism)d(of)i Fp(K)17 b Fx(-algebras)28 b Ft(g)j Fx(:)c Ft(T)14 b Fx(\()p Ft(V)22 b Fx(\))27 b Fu(\000)-60 b(!)28 b Ft(A)0 2774 y Fx(suc)m(h)34 b(that)e(the)h (diagram)1480 2911 y Ft(V)358 b(T)14 b Fx(\()p Ft(V)21 b Fx(\))p 1588 2888 278 4 v 1783 2886 a Fl(-)1709 2868 y Ft(\023)1644 3155 y(f)1588 3038 y Fl(@)1671 3121 y(@)1754 3204 y(@)1837 3287 y(@)1856 3306 y(@)-83 b(R)1971 3408 y Ft(A)p 2006 3306 4 351 v 2007 3306 a Fl(?)2046 3142 y Ft(g)0 3535 y Fx(comm)m(utes.)149 3651 y Fs(Note:)42 b Fx(If)27 b(y)m(ou)i(w)m(an)m(t)g(to)e(de\014ne)i(a)f(homomorphism)d Ft(g)31 b Fx(:)c Ft(T)14 b Fx(\()p Ft(V)22 b Fx(\))27 b Fu(\000)-60 b(!)28 b Ft(A)g Fx(with)f(a)h(tensor)g(algebra)0 3767 y(as)34 b(domain)f(y)m(ou)i(should)f(de\014ne)i(it)d(b)m(y)i (giving)e(a)h(homomorphism)d(of)j Fp(K)17 b Fx(-mo)s(dules)39 b(de\014ned)d(on)0 3883 y Ft(V)22 b Fx(.)149 4071 y Fv(Lemma)38 b(8.5.5.)49 b Fs(A)34 b(tensor)g(algebr)-5 b(a)33 b Fx(\()p Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))p Ft(;)c(\023)p Fx(\))34 b Fs(de\014ne)-5 b(d)33 b(by)h Ft(V)56 b Fs(is)33 b(unique)h(up)g(to)g (a)g(unique)0 4188 y(isomorphism.)149 4376 y Fi(Pr)n(oof.)42 b Fx(Let)33 b(\()p Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))p Ft(;)c(\023)p Fx(\))33 b(and)g(\()p Ft(T)1420 4340 y Fr(0)1443 4376 y Fx(\()p Ft(V)21 b Fx(\))p Ft(;)c(\023)1675 4340 y Fr(0)1698 4376 y Fx(\))33 b(b)s(e)g(tensor)g(algebras)f(o)m(v)m (er)h Ft(V)22 b Fx(.)43 b(Then)1755 4541 y Ft(V)1210 4772 y(\023)1575 4617 y Fl(\010)1492 4658 y(\010)1409 4700 y(\010)1326 4741 y(\010)1243 4783 y(\010)1160 4824 y(\010)1077 4866 y(\010)993 4907 y(\010)956 4926 y(\010)-83 b(\031)1611 4787 y Ft(\023)1645 4751 y Fr(0)1643 4658 y Fl(\000)1560 4741 y(\000)1477 4824 y(\000)1394 4907 y(\000)1375 4926 y(\000)g(\011)1943 4772 y Ft(\023)1862 4658 y Fl(@)1945 4741 y(@)2028 4824 y(@)2111 4907 y(@)2130 4926 y(@)g(R)2344 4787 y Ft(\023)2378 4751 y Fr(0)1931 4617 y Fl(H)2014 4658 y(H)2097 4700 y(H)2180 4741 y(H)2263 4783 y(H)2346 4824 y(H)2429 4866 y(H)2512 4907 y(H)2550 4926 y(H)g(j)707 5019 y Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))250 b Ft(T)1253 4983 y Fr(0)1276 5019 y Fx(\()p Ft(V)22 b Fx(\))p 961 4996 194 4 v 1072 4994 a Fl(-)1029 4975 y Ft(h)p 1460 4996 681 4 v 2058 4994 a Fl(-)1773 4975 y Ft(k)2169 5019 y(T)14 b Fx(\()p Ft(V)21 b Fx(\))251 b Ft(T)2716 4983 y Fr(0)2739 5019 y Fx(\()p Ft(V)21 b Fx(\))p 2423 4996 194 4 v 2534 4994 a Fl(-)2491 4975 y Ft(h)0 5196 y Fx(implies)30 b Ft(k)h Fx(=)c Ft(h)572 5160 y Fr(\000)p Fn(1)667 5196 y Fx(.)p 3529 5196 4 66 v 3533 5134 59 4 v 3533 5196 V 3591 5196 4 66 v eop %%Page: 19 19 19 18 bop 1530 236 a Fo(5.)77 b(ALGEBRAS)1427 b(19)149 425 y Fv(Prop)s(osition)36 b(8.5.6.)49 b Fx(\(Rules)34 b(of)g(computation)e(in)h(a)h(tensor)g(algebra\))h Fs(L)-5 b(et)37 b Fx(\()p Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))p Ft(;)c(\023)p Fx(\))36 b Fs(b)-5 b(e)0 541 y(the)35 b(tensor)g(algebr) -5 b(a)34 b(over)g Ft(V)22 b Fs(.)44 b(Then)34 b(we)h(have)149 680 y Fx(1.)42 b Ft(\023)32 b Fx(:)g Ft(V)53 b Fu(\000)-58 b(!)32 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))37 b Fs(is)g(inje)-5 b(ctive)36 b(\(so)h(we)f(may)h(identify)g(the)g(elements)f Ft(\023)p Fx(\()p Ft(v)t Fx(\))h Fs(and)g Ft(v)k Fs(for)36 b(al)5 b(l)267 796 y Ft(v)31 b Fu(2)d Ft(V)22 b Fs(\),)149 912 y Fx(2.)42 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))28 b(=)f Fu(f)673 837 y Ff(P)778 941 y Fq(n;)835 926 y Fn(\026)841 941 y Fq(i)885 912 y Ft(v)932 927 y Fq(i)956 936 y Fc(1)1017 912 y Fu(\001)22 b Ft(:)17 b(:)g(:)22 b Fu(\001)g Ft(v)1301 927 y Fq(i)1325 935 y Fk(n)1372 912 y Fu(j)1392 890 y Fx(\026)1400 912 y Ft(i)27 b Fx(=)h(\()p Ft(i)1635 927 y Fn(1)1674 912 y Ft(;)17 b(:)g(:)g(:)33 b(;)17 b(i)1943 927 y Fq(n)1990 912 y Fx(\))35 b Fs(multiindex)f(of)g(length)h Ft(n)p Fu(g)p Ft(;)149 1032 y Fx(3.)42 b Fs(if)g Ft(f)53 b Fx(:)43 b Ft(V)63 b Fu(\000)-58 b(!)43 b Ft(A)g Fs(is)g(a)f (homomorphism)f(of)h Fp(K)18 b Fs(-mo)-5 b(dules,)49 b Ft(A)43 b Fs(is)g(a)f Fp(K)18 b Fs(-algebr)-5 b(a,)49 b(and)42 b Ft(g)k Fx(:)267 1148 y Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))28 b Fu(\000)-58 b(!)28 b Ft(A)35 b Fs(is)f(the)h(induc)-5 b(e)g(d)35 b(homomorphism)d(of)j Fp(K)17 b Fs(-algebr)-5 b(as,)40 b(then)876 1329 y Ft(g)t Fx(\()965 1235 y Ff(X)993 1454 y Fq(n;)1050 1439 y Fn(\026)1056 1454 y Fq(i)1125 1329 y Ft(v)1172 1344 y Fq(i)1196 1353 y Fc(1)1257 1329 y Fu(\001)22 b Ft(:)17 b(:)g(:)22 b Fu(\001)g Ft(v)1541 1344 y Fq(i)1565 1352 y Fk(n)1611 1329 y Fx(\))28 b(=)1781 1235 y Ff(X)1809 1454 y Fq(n;)1866 1439 y Fn(\026)1872 1454 y Fq(i)1941 1329 y Ft(f)11 b Fx(\()p Ft(v)2085 1344 y Fq(i)2109 1353 y Fc(1)2148 1329 y Fx(\))22 b Fu(\001)g Ft(:)17 b(:)g(:)k Fu(\001)h Ft(f)11 b Fx(\()p Ft(v)2588 1344 y Fq(i)2612 1352 y Fk(n)2658 1329 y Fx(\))p Ft(:)149 1620 y Fi(Pr)n(oof.)42 b Fx(1.)64 b(Use)40 b(the)f(em)m(b)s(edding)g (homomorphism)e Ft(j)45 b Fx(:)39 b Ft(V)60 b Fu(\000)-60 b(!)40 b Ft(D)s Fx(\()p Ft(V)21 b Fx(\),)41 b(where)f Ft(D)s Fx(\()p Ft(V)21 b Fx(\))39 b(is)0 1736 y(de\014ned)h(as)e(in)f (8.5.3.)60 b(to)38 b(construct)h Ft(g)i Fx(:)c Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))37 b Fu(\000)-60 b(!)37 b Ft(D)s Fx(\()p Ft(V)22 b Fx(\))38 b(suc)m(h)h(that)f Ft(g)29 b Fu(\016)d Ft(\023)38 b Fx(=)f Ft(j)6 b Fx(.)60 b(Since)38 b Ft(j)44 b Fx(is)0 1853 y(injectiv)m(e)32 b(so)h(is)f Ft(\023)p Fx(.)149 1969 y(2.)43 b(Let)31 b Ft(B)i Fx(:=)27 b Fu(f)728 1894 y Ff(P)833 1998 y Fq(n;)890 1982 y Fn(\026)896 1998 y Fq(i)940 1969 y Ft(v)987 1984 y Fq(i)1011 1993 y Fc(1)1069 1969 y Fu(\001)17 b Ft(:)g(:)g(:)h Fu(\001)g Ft(v)1340 1984 y Fq(i)1364 1992 y Fk(n)1411 1969 y Fu(j)1431 1946 y Fx(\026)1439 1969 y Ft(i)27 b Fx(=)h(\()p Ft(i)1674 1984 y Fn(1)1713 1969 y Ft(;)17 b(:)g(:)g(:)33 b(;)17 b(i)1982 1984 y Fq(n)2029 1969 y Fx(\))32 b(m)m(ultiindex)f(of)h (length)g Ft(n)p Fu(g)p Fx(.)43 b(Ob)m(viously)0 2094 y Ft(B)d Fx(is)34 b(the)h(subalgebra)f(of)g Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))35 b(generated)g(b)m(y)h(the)f(elemen)m(ts)g (of)f Ft(V)21 b Fx(.)50 b(Let)34 b Ft(j)k Fx(:)31 b Ft(B)36 b Fu(\000)-60 b(!)31 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))35 b(b)s(e)0 2210 y(the)k(em)m(b)s(edding)f(homomorphism.)60 b(Then)39 b Ft(\023)g Fx(:)f Ft(V)60 b Fu(\000)-60 b(!)38 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))39 b(factors)g(through)f(a)h (linear)e(map)0 2326 y Ft(\023)34 2290 y Fr(0)85 2326 y Fx(:)28 b Ft(V)49 b Fu(\000)-60 b(!)28 b Ft(B)5 b Fx(.)44 b(In)33 b(the)g(follo)m(wing)d(diagram)1226 2535 y Ft(V)430 b(B)p 1333 2503 351 4 v 1601 2501 a Fl(-)1480 2482 y Ft(\023)1514 2446 y Fr(0)2128 2526 y Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))p 1821 2503 278 4 v 2016 2501 a Fl(-)1936 2463 y Ft(j)1713 3013 y(B)341 b(T)14 b Fx(\()p Ft(V)21 b Fx(\))p 1821 2991 V 2016 2989 a Fl(-)1936 2950 y Ft(j)1391 2781 y(\023)1425 2745 y Fr(0)1333 2652 y Fl(@)1417 2735 y(@)1500 2818 y(@)1583 2901 y(@)1601 2920 y(@)-83 b(R)p 1751 2920 4 351 v 69 w(?)1572 2772 y Fx(id)1653 2787 y Fq(B)p 2239 2920 V 2240 2920 a Fl(?)2279 2768 y Ft(j)6 b(p)2057 2757 y(p)2089 2652 y Fl(\000)2006 2735 y(\000)1923 2818 y(\000)1840 2901 y(\000)1821 2920 y(\000)-83 b(\011)0 3163 y Fx(w)m(e)37 b(ha)m(v)m(e)h(id)457 3178 y Fq(B)534 3163 y Fu(\016)p Ft(\023)618 3127 y Fr(0)675 3163 y Fx(=)c Ft(\023)819 3127 y Fr(0)843 3163 y Fx(.)54 b Ft(p)36 b Fx(with)g Ft(p)24 b Fu(\016)h Ft(j)30 b Fu(\016)24 b Ft(\023)1561 3127 y Fr(0)1619 3163 y Fx(=)33 b Ft(p)25 b Fu(\016)f Ft(\023)35 b Fx(=)e Ft(\023)2088 3127 y Fr(0)2148 3163 y Fx(exists)k(since)f Ft(\023)2697 3127 y Fr(0)2757 3163 y Fx(is)g(a)g(homomorphism)0 3280 y(of)c Fp(K)17 b Fx(-mo)s(dules.)48 b(Because)34 b(of)e Ft(j)6 b(p)22 b Fu(\016)f Ft(\023)29 b Fx(=)e Ft(j)h Fu(\016)21 b Ft(\023)1652 3244 y Fr(0)1704 3280 y Fx(=)27 b Ft(\023)i Fx(=)e(id)2054 3295 y Fq(T)10 b Fn(\()p Fq(V)17 b Fn(\))2238 3280 y Fu(\016)p Ft(\023)32 b Fx(w)m(e)i(get)e Ft(j)6 b(p)28 b Fx(=)f(id)2968 3295 y Fq(T)10 b Fn(\()p Fq(V)16 b Fn(\))3134 3280 y Fx(,)33 b(hence)h(the)0 3396 y(em)m(b)s(edding)e Ft(j)39 b Fx(is)32 b(surjectiv)m(e)i(and)e(th)m(us)i Ft(j)k Fx(is)32 b(the)h(iden)m(tit)m(y)-8 b(.)149 3512 y(3.)44 b(is)32 b(precisely)h(the)g(de\014nition)e(of)h(the)h(induced)g (homomorphism.)p 3529 3512 4 66 v 3533 3450 59 4 v 3533 3512 V 3591 3512 4 66 v 149 3688 a Fv(Prop)s(osition)j(8.5.7.)49 b Fs(Given)d(a)h Fp(K)17 b Fs(-mo)-5 b(dule)52 b Ft(V)22 b Fs(.)79 b(Then)46 b(ther)-5 b(e)46 b(exists)g(a)h(tensor)f(algebr)-5 b(a)0 3805 y Fx(\()p Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))p Ft(;)c(\023)p Fx(\))p Fs(.)149 3981 y Fi(Pr)n(oof.)42 b Fx(De\014ne)35 b Ft(T)909 3945 y Fq(n)956 3981 y Fx(\()p Ft(V)22 b Fx(\))30 b(:=)h Ft(V)45 b Fu(\012)24 b Ft(:)17 b(:)g(:)23 b Fu(\012)h Ft(V)52 b Fx(=)31 b Ft(V)2012 3945 y Fr(\012)p Fq(n)2148 3981 y Fx(to)j(b)s(e)h(the)g Ft(n)p Fx(-fold)e(tensor)i(pro)s(duct)f(of)0 4097 y Ft(V)22 b Fx(.)43 b(De\014ne)33 b Ft(T)522 4061 y Fn(0)561 4097 y Fx(\()p Ft(V)22 b Fx(\))27 b(:=)h Fp(K)56 b Fx(and)33 b Ft(T)1245 4061 y Fn(1)1284 4097 y Fx(\()p Ft(V)21 b Fx(\))28 b(:=)g Ft(V)21 b Fx(.)44 b(W)-8 b(e)33 b(de\014ne)469 4279 y Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))27 b(:=)852 4184 y Ff(M)870 4394 y Fq(i)p Fr(\025)p Fn(0)1019 4279 y Ft(T)1090 4237 y Fq(i)1118 4279 y Fx(\()p Ft(V)22 b Fx(\))27 b(=)h Fp(K)45 b Fu(\010)23 b Ft(V)44 b Fu(\010)22 b Fx(\()p Ft(V)44 b Fu(\012)23 b Ft(V)e Fx(\))h Fu(\010)h Fx(\()p Ft(V)43 b Fu(\012)23 b Ft(V)44 b Fu(\012)22 b Ft(V)g Fx(\))g Fu(\010)h Ft(:)17 b(:)g(:)32 b(:)0 4542 y Fx(The)i(comp)s(onen)m(ts)f Ft(T)809 4506 y Fq(n)855 4542 y Fx(\()p Ft(V)22 b Fx(\))32 b(of)g Ft(T)14 b Fx(\()p Ft(V)22 b Fx(\))32 b(are)h(called)e Fs(homo)-5 b(gene)g(ous)33 b(c)-5 b(omp)g(onents)p Fx(.)149 4659 y(The)33 b(canonical)e (isomorphisms)f Ft(T)1456 4623 y Fq(m)1522 4659 y Fx(\()p Ft(V)22 b Fx(\))f Fu(\012)g Ft(T)1867 4623 y Fq(n)1914 4659 y Fx(\()p Ft(V)g Fx(\))2096 4631 y Fu(\030)2097 4663 y Fx(=)2201 4659 y Ft(T)2272 4623 y Fq(m)p Fn(+)p Fq(n)2436 4659 y Fx(\()p Ft(V)h Fx(\))31 b(tak)m(en)i(as)g(m)m (ultiplication)1094 4817 y Fu(r)27 b Fx(:)h Ft(T)1330 4781 y Fq(m)1396 4817 y Fx(\()p Ft(V)22 b Fx(\))g Fu(\012)h Ft(T)1744 4781 y Fq(n)1790 4817 y Fx(\()p Ft(V)f Fx(\))27 b Fu(\000)-60 b(!)28 b Ft(T)2188 4781 y Fq(m)p Fn(+)p Fq(n)2352 4817 y Fx(\()p Ft(V)22 b Fx(\))1232 4933 y Fu(r)28 b Fx(:)g Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))h Fu(\012)h Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))28 b Fu(\000)-60 b(!)27 b Ft(T)14 b Fx(\()p Ft(V)22 b Fx(\))0 5099 y(and)36 b(the)h(em)m(b)s(edding)e Ft(\021)j Fx(:)33 b Fp(K)57 b Fx(=)34 b Ft(T)1299 5063 y Fn(0)1338 5099 y Fx(\()p Ft(V)21 b Fx(\))34 b Fu(\000)-60 b(!)34 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))36 b(induce)g(the)g(structure)i(of)d(a)h Fp(K)18 b Fx(-algebra)40 b(on)0 5216 y Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\).)44 b(F)-8 b(urthermore)32 b(w)m(e)h(ha)m(v)m(e)h(the)f (em)m(b)s(edding)f Ft(\023)c Fx(:)g Ft(V)49 b Fu(\000)-60 b(!)28 b Ft(T)2329 5179 y Fn(1)2368 5216 y Fx(\()p Ft(V)22 b Fx(\))27 b Fu(\022)i Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\).)p eop %%Page: 20 20 20 19 bop 0 236 a Fo(20)1456 b(8.)77 b(TOOLBO)n(X)149 425 y Fx(W)-8 b(e)42 b(ha)m(v)m(e)h(to)e(sho)m(w)h(that)f(\()p Ft(T)14 b Fx(\()p Ft(V)22 b Fx(\))p Ft(;)17 b(\023)p Fx(\))41 b(is)g(a)g(tensor)h(algebra.)68 b(Let)42 b Ft(f)53 b Fx(:)43 b Ft(V)64 b Fu(\000)-60 b(!)43 b Ft(A)e Fx(b)s(e)g(a)h(ho-)0 541 y(momorphism)c(of)i Fp(K)17 b Fx(-mo)s(dules.)73 b(Eac)m(h)41 b(elemen)m(t)f(in)g Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))41 b(is)f(a)g(sum)h(of)f(decomp)s(osable)g(ten-)0 658 y(sors)46 b Ft(v)256 673 y Fn(1)327 658 y Fu(\012)32 b Ft(:)17 b(:)g(:)30 b Fu(\012)i Ft(v)737 673 y Fq(n)784 658 y Fx(.)83 b(De\014ne)47 b Ft(g)53 b Fx(:)e Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))50 b Fu(\000)-60 b(!)50 b Ft(A)c Fx(b)m(y)h Ft(g)t Fx(\()p Ft(v)2234 673 y Fn(1)2304 658 y Fu(\012)32 b Ft(:)17 b(:)g(:)30 b Fu(\012)i Ft(v)2714 673 y Fq(n)2761 658 y Fx(\))50 b(:=)g Ft(f)11 b Fx(\()p Ft(v)3146 673 y Fn(1)3186 658 y Fx(\))17 b Ft(:)g(:)g(:)e(f)c Fx(\()p Ft(v)3515 673 y Fq(n)3562 658 y Fx(\))0 774 y(\(and)50 b(\()p Ft(g)60 b Fx(:)c Ft(T)544 738 y Fn(0)584 774 y Fx(\()p Ft(V)21 b Fx(\))57 b Fu(\000)-60 b(!)57 b Ft(A)p Fx(\))f(=)h(\()p Ft(\021)j Fx(:)d Fp(K)80 b Fu(\000)-60 b(!)57 b Ft(A)p Fx(\)\).)95 b(By)50 b(induction)f(one)h(sees)h(that)e Ft(g)k Fx(is)c(a)0 890 y(homomorphism)e(of)i(algebras.)95 b(Since)50 b(\()p Ft(g)60 b Fx(:)d Ft(T)1876 854 y Fn(1)1915 890 y Fx(\()p Ft(V)21 b Fx(\))57 b Fu(\000)-60 b(!)57 b Ft(A)p Fx(\))g(=)g(\()p Ft(f)68 b Fx(:)57 b Ft(V)78 b Fu(\000)-60 b(!)57 b Ft(A)p Fx(\))50 b(w)m(e)h(get)0 1006 y Ft(g)33 b Fu(\016)d Ft(\023)47 b Fx(=)g Ft(f)11 b Fx(.)77 b(If)44 b Ft(h)j Fx(:)h Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))47 b Fu(\000)-60 b(!)47 b Ft(A)d Fx(is)f(a)h(homomorphism)d(of) i(algebras)h(with)f Ft(h)30 b Fu(\016)g Ft(\023)47 b Fx(=)g Ft(f)55 b Fx(w)m(e)0 1123 y(get)33 b Ft(h)p Fx(\()p Ft(v)304 1138 y Fn(1)365 1123 y Fu(\012)23 b Ft(:)17 b(:)g(:)22 b Fu(\012)g Ft(v)748 1138 y Fq(n)795 1123 y Fx(\))28 b(=)g Ft(h)p Fx(\()p Ft(v)1106 1138 y Fn(1)1145 1123 y Fx(\))17 b Ft(:)g(:)g(:)f(h)p Fx(\()p Ft(v)1472 1138 y Fq(n)1519 1123 y Fx(\))28 b(=)f Ft(f)11 b Fx(\()p Ft(v)1832 1138 y Fn(1)1872 1123 y Fx(\))17 b Ft(:)g(:)g(:)e(f)c Fx(\()p Ft(v)2201 1138 y Fq(n)2248 1123 y Fx(\))32 b(hence)i Ft(h)28 b Fx(=)g Ft(g)t Fx(.)p 3529 1123 4 66 v 3533 1060 59 4 v 3533 1123 V 3591 1123 4 66 v 149 1298 a Fv(Prop)s(osition) 36 b(8.5.8.)49 b Fs(The)37 b(c)-5 b(onstruction)38 b(of)f(tensor)h (algebr)-5 b(as)37 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))38 b Fs(de\014nes)e(a)i(functor)0 1414 y Ft(T)52 b Fx(:)38 b Fp(K)17 b Fs(-)6 b Fv(Mo)s(d)39 b Fu(\000)-58 b(!)38 b Fp(K)18 b Fs(-)6 b Fv(Alg)39 b Fs(that)i(is)f(left)g(adjoint)g(to)h (the)f(underlying)g(functor)h Ft(U)48 b Fx(:)38 b Fp(K)18 b Fs(-)6 b Fv(Alg)36 b Fu(\000)-58 b(!)0 1530 y Fp(K)17 b Fs(-)6 b Fv(Mo)s(d)q Fs(.)149 1706 y Fi(Pr)n(oof.)42 b Fx(F)-8 b(ollo)m(ws)31 b(from)h(the)h(univ)m(ersal)f(prop)s(ert)m(y)i (and)e(8.9.16.)p 3529 1706 V 3533 1643 59 4 v 3533 1706 V 3591 1706 4 66 v 149 1881 a Fv(Problem)k(8.5.6.)49 b Fx(1.)71 b(Let)43 b Ft(X)49 b Fx(b)s(e)42 b(a)g(set)g(and)g Ft(V)65 b Fx(:=)44 b Fp(K)17 b Ft(X)56 b Fx(b)s(e)42 b(the)g(free)g Fp(K)18 b Fx(-mo)s(dule)46 b(o)m(v)m(er)0 1997 y Ft(X)8 b Fx(.)74 b(Sho)m(w)44 b(that)e Ft(X)53 b Fu(\000)-60 b(!)45 b Ft(V)67 b Fu(\000)-60 b(!)45 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))43 b(de\014nes)i(a)d Fs(fr)-5 b(e)g(e)44 b(algebr)-5 b(a)42 b Fx(o)m(v)m(er)i Ft(X)8 b Fx(,)45 b(i.e.)74 b(for)42 b(ev)m(ery)j Fp(K)17 b Fx(-)0 2114 y(algebra)30 b Ft(A)g Fx(and)h(ev)m(ery)i(map)d Ft(f)38 b Fx(:)28 b Ft(X)35 b Fu(\000)-60 b(!)28 b Ft(A)j Fx(there)g(is)g(a)f(unique)h(homomorphism)d(of)j Fp(K)17 b Fx(-algebras)0 2230 y Ft(g)31 b Fx(:)d Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))28 b Fu(\000)-60 b(!)27 b Ft(A)33 b Fx(suc)m(h)h(that)e (the)h(diagram)1478 2380 y Ft(X)338 b(T)14 b Fx(\()p Ft(V)21 b Fx(\))p 1595 2357 273 4 v 1785 2355 a Fl(-)1647 2624 y Ft(f)1590 2507 y Fl(@)1673 2590 y(@)1756 2673 y(@)1839 2756 y(@)1858 2775 y(@)-83 b(R)1973 2877 y Ft(A)p 2008 2775 4 351 v 2010 2775 a Fl(?)2049 2611 y Ft(g)0 3002 y Fx(comm)m(utes.)149 3118 y(W)-8 b(e)43 b(write)f Fp(K)17 b Fu(h)p Ft(X)8 b Fu(i)50 b Fx(:=)43 b Ft(T)14 b Fx(\()p Fp(K)k Ft(X)8 b Fx(\))47 b(and)c(call)d(it)h(the)i Fs(p)-5 b(olynomial)42 b(ring)h(over)h Fp(K)67 b Fs(in)43 b(the)h(non-)0 3234 y(c)-5 b(ommuting)34 b(variables)e Ft(X)8 b Fx(.)149 3350 y(2.)64 b(Let)40 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))39 b(and)h Ft(\023)g Fx(:)f Ft(V)61 b Fu(\000)-60 b(!)39 b Ft(T)14 b Fx(\()p Ft(V)22 b Fx(\))39 b(b)s(e)g(a)h(tensor)g(algebra.)63 b(Regard)39 b Ft(V)61 b Fx(as)39 b(a)g(subset)i(of)0 3467 y Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))36 b(b)m(y)h Ft(\023)p Fx(.)53 b(Sho)m(w)37 b(that)f(there)g(is)f(a)h(unique)g(homomorphism)d(\001)h(:)f Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))33 b Fu(\000)-60 b(!)34 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))j Fu(\012)h Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))0 3583 y(with)32 b(\001\()p Ft(v)t Fx(\))c(=)f Ft(v)f Fu(\012)d Fx(1)f(+)g(1)g Fu(\012)g Ft(v)37 b Fx(for)32 b(all)e Ft(v)h Fu(2)e Ft(V)21 b Fx(.)149 3699 y(3.)44 b(Sho)m(w)33 b(that)f(\(\001)23 b Fu(\012)f Fx(1\)\001)28 b(=)g(\(1)22 b Fu(\012)g Fx(\001\)\001)28 b(:)g Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))28 b Fu(\000)-60 b(!)28 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))h Fu(\012)h Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))h Fu(\012)g Ft(T)14 b Fx(\()p Ft(V)22 b Fx(\).)149 3815 y(4.)61 b(Sho)m(w)39 b(that)f(there)h(is)e(a)h(unique)h(homomorphism)c(of)j(algebras)g Ft(")f Fx(:)g Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))37 b Fu(\000)-60 b(!)38 b Fp(K)61 b Fx(with)0 3931 y Ft(")p Fx(\()p Ft(v)t Fx(\))27 b(=)g(0)33 b(for)f(all)e Ft(v)i Fu(2)c Ft(V)21 b Fx(.)149 4048 y(5.)44 b(Sho)m(w)33 b(that)f(\()p Ft(")22 b Fu(\012)h Fx(1\)\001)k(=)h(\(1)22 b Fu(\012)g Ft(")p Fx(\)\001)28 b(=)g(id)1828 4063 y Fq(T)10 b Fn(\()p Fq(V)16 b Fn(\))1995 4048 y Fx(.)149 4164 y(6.)58 b(Sho)m(w)39 b(that)e(there)h(is)f(a)g(unique)h(homomorphism)d(of)i(algebras)g Ft(S)42 b Fx(:)36 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))36 b Fu(\000)-60 b(!)36 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))3526 4128 y Fq(op)0 4280 y Fx(with)j Ft(S)6 b Fx(\()p Ft(v)t Fx(\))27 b(=)h Fu(\000)p Ft(v)t Fx(.)41 b(\()p Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))997 4244 y Fq(op)1095 4280 y Fx(is)j(the)h Fs(opp)-5 b(osite)27 b(algebr)-5 b(a)23 b Fx(of)h Ft(T)14 b Fx(\()p Ft(V)22 b Fx(\))i(with)g(m)m(ultiplication) c Ft(s)6 b Fu(\003)g Ft(t)27 b Fx(:=)h Ft(ts)0 4396 y Fx(for)k(all)f Ft(s;)17 b(t)27 b Fu(2)h Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))28 b(=)g Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))1113 4360 y Fq(op)1219 4396 y Fx(and)33 b(where)g Ft(st)g Fx(denotes)h(the)f(pro)s(duct)g(in)f Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\).\))149 4513 y(7.)44 b(Sho)m(w)33 b(that)f(the)h(diagrams)1297 4668 y Ft(T)14 b Fx(\()p Ft(V)22 b Fx(\))238 b Fp(K)p 1551 4645 182 4 v 1650 4643 a Fl(-)1626 4624 y Fq(")2077 4668 y Ft(T)14 b Fx(\()p Ft(V)22 b Fx(\))p 1867 4645 V 1966 4643 a Fl(-)1939 4611 y Fq(\021)1124 5058 y Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))h Fu(\012)h Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))208 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))h Fu(\012)h Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))p 1725 5035 151 4 v 1793 5033 a Fl(-)1732 5008 y Fn(1)p Fr(\012)p Fq(S)1732 5097 y(S)t Fr(\012)p Fn(1)p 1408 4965 4 254 v 1410 4965 a Fl(?)1312 4861 y Fn(\001)p 2188 4965 V 2190 4795 a Fl(6)2229 4861 y Fr(r)0 5216 y Fx(comm)m(ute.)p eop %%Page: 21 21 21 20 bop 1530 236 a Fo(5.)77 b(ALGEBRAS)1427 b(21)149 425 y Fv(De\014nition)37 b(8.5.9.)49 b Fx(Let)38 b Fp(K)62 b Fx(b)s(e)38 b(a)g(comm)m(utativ)m(e)f(ring.)60 b(Let)38 b Ft(V)60 b Fx(b)s(e)38 b(a)g Fp(K)17 b Fx(-mo)s(dule.)65 b(A)38 b Fp(K)17 b Fx(-)0 541 y(algebra)29 b Ft(S)6 b Fx(\()p Ft(V)22 b Fx(\))30 b(together)g(with)g(a)g(homomorphism)e(of)i Fp(K)17 b Fx(-mo)s(dules)35 b Ft(\023)29 b Fx(:)e Ft(V)50 b Fu(\000)-60 b(!)27 b Ft(S)6 b Fx(\()p Ft(V)22 b Fx(\),)30 b(suc)m(h)i(that)0 658 y Ft(\023)p Fx(\()p Ft(v)t Fx(\))25 b Fu(\001)g Ft(\023)p Fx(\()p Ft(v)362 622 y Fr(0)386 658 y Fx(\))35 b(=)g Ft(\023)p Fx(\()p Ft(v)693 622 y Fr(0)717 658 y Fx(\))25 b Fu(\001)g Ft(\023)p Fx(\()p Ft(v)t Fx(\))37 b(for)f(all)g Ft(v)t(;)17 b(v)1471 622 y Fr(0)1528 658 y Fu(2)36 b Ft(V)22 b Fx(,)38 b(is)f(called)f(a)g Fs(symmetric)j(algebr)-5 b(a)38 b(over)f Ft(V)59 b Fx(if)35 b(for)0 774 y(eac)m(h)h Fp(K)18 b Fx(-algebra)39 b Ft(A)d Fx(and)f(for)g(eac)m(h)h(homomorphism)d(of)h Fp(K)18 b Fx(-mo)s(dules)40 b Ft(f)j Fx(:)32 b Ft(V)54 b Fu(\000)-60 b(!)32 b Ft(A)p Fx(,)k(suc)m(h)h(that)0 890 y Ft(f)11 b Fx(\()p Ft(v)t Fx(\))25 b Fu(\001)h Ft(f)11 b Fx(\()p Ft(v)413 854 y Fr(0)436 890 y Fx(\))37 b(=)g Ft(f)11 b Fx(\()p Ft(v)772 854 y Fr(0)795 890 y Fx(\))26 b Fu(\001)g Ft(f)11 b Fx(\()p Ft(v)t Fx(\))37 b(for)h(all)e Ft(v)t(;)17 b(v)1578 854 y Fr(0)1638 890 y Fu(2)38 b Ft(V)21 b Fx(,)40 b(there)f(exists)g(a)f(unique)h(homomorphism)c(of)0 1006 y Fp(K)17 b Fx(-algebras)38 b Ft(g)31 b Fx(:)d Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))28 b Fu(\000)-60 b(!)27 b Ft(A)33 b Fx(suc)m(h)h(that)e(the)h(diagram)1482 1183 y Ft(V)360 b(S)6 b Fx(\()p Ft(V)21 b Fx(\))p 1589 1159 281 4 v 1787 1157 a Fl(-)1713 1139 y Ft(\023)1646 1426 y(f)1589 1309 y Fl(@)1672 1392 y(@)1755 1475 y(@)1838 1558 y(@)1857 1577 y(@)-83 b(R)1972 1679 y Ft(A)p 2007 1577 4 351 v 2009 1577 a Fl(?)2048 1413 y Ft(g)0 1819 y Fx(comm)m(utes.)149 1936 y Fs(Note:)56 b Fx(If)38 b(y)m(ou)h(w)m(an)m(t)h(to)e(de\014ne)h (a)f(homomorphism)e Ft(g)41 b Fx(:)d Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))37 b Fu(\000)-60 b(!)38 b Ft(A)h Fx(with)f(a)g (symmetric)0 2052 y(algebra)44 b(as)i(domain)d(y)m(ou)j(should)f (de\014ne)i(it)d(b)m(y)i(giving)e(a)h(homomorphism)d(of)j Fp(K)17 b Fx(-mo)s(dules)0 2168 y Ft(f)38 b Fx(:)28 b Ft(V)49 b Fu(\000)-60 b(!)28 b Ft(A)33 b Fx(satisfying)e Ft(f)11 b Fx(\()p Ft(v)t Fx(\))22 b Fu(\001)g Ft(f)11 b Fx(\()p Ft(v)1338 2132 y Fr(0)1360 2168 y Fx(\))28 b(=)g Ft(f)11 b Fx(\()p Ft(v)1678 2132 y Fr(0)1700 2168 y Fx(\))22 b Fu(\001)g Ft(f)11 b Fx(\()p Ft(v)t Fx(\))32 b(for)g(all)e Ft(v)t(;)17 b(v)2458 2132 y Fr(0)2509 2168 y Fu(2)28 b Ft(V)21 b Fx(.)149 2352 y Fv(Lemma)38 b(8.5.10.)49 b Fs(A)42 b(symmetric)f(algebr)-5 b(a)41 b Fx(\()p Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))p Ft(;)c(\023)p Fx(\))42 b Fs(de\014ne)-5 b(d)41 b(by)g Ft(V)64 b Fs(is)41 b(unique)h(up)f(to)h(a) 0 2468 y(unique)35 b(isomorphism.)149 2653 y Fi(Pr)n(oof.)42 b Fx(Let)33 b(\()p Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))p Ft(;)c(\023)p Fx(\))33 b(and)g(\()p Ft(S)1410 2616 y Fr(0)1433 2653 y Fx(\()p Ft(V)21 b Fx(\))p Ft(;)c(\023)1665 2616 y Fr(0)1689 2653 y Fx(\))32 b(b)s(e)h(symmetric)f(algebras)f(o)m (v)m(er)j Ft(V)22 b Fx(.)43 b(Then)1755 2811 y Ft(V)1210 3042 y(\023)1575 2886 y Fl(\010)1492 2928 y(\010)1409 2969 y(\010)1326 3011 y(\010)1243 3053 y(\010)1160 3094 y(\010)1077 3136 y(\010)993 3177 y(\010)956 3196 y(\010)-83 b(\031)1611 3057 y Ft(\023)1645 3021 y Fr(0)1643 2928 y Fl(\000)1560 3011 y(\000)1477 3094 y(\000)1394 3177 y(\000)1375 3196 y(\000)g(\011)1943 3042 y Ft(\023)1862 2928 y Fl(@)1945 3011 y(@)2028 3094 y(@)2111 3177 y(@)2130 3196 y(@)g(R)2344 3057 y Ft(\023)2378 3021 y Fr(0)1931 2886 y Fl(H)2014 2928 y(H)2097 2969 y(H)2180 3011 y(H)2263 3053 y(H)2346 3094 y(H)2429 3136 y(H)2512 3177 y(H)2550 3196 y(H)g(j)709 3289 y Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))256 b Ft(S)1251 3253 y Fr(0)1274 3289 y Fx(\()p Ft(V)21 b Fx(\))p 958 3266 199 4 v 1074 3264 a Fl(-)1029 3245 y Ft(h)p 1457 3266 687 4 v 2061 3264 a Fl(-)1773 3245 y Ft(k)2172 3289 y(S)6 b Fx(\()p Ft(V)21 b Fx(\))255 b Ft(S)2713 3253 y Fr(0)2736 3289 y Fx(\()p Ft(V)22 b Fx(\))p 2420 3266 199 4 v 2536 3264 a Fl(-)2491 3245 y Ft(h)0 3459 y Fx(implies)30 b Ft(k)h Fx(=)c Ft(h)572 3423 y Fr(\000)p Fn(1)667 3459 y Fx(.)p 3529 3459 4 66 v 3533 3396 59 4 v 3533 3459 V 3591 3459 4 66 v 149 3643 a Fv(Prop)s(osition) 36 b(8.5.11.)49 b Fx(\(Rules)22 b(of)f(computation)f(in)h(a)h (symmetric)f(algebra\))i Fs(L)-5 b(et)26 b Fx(\()p Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))p Ft(;)c(\023)p Fx(\))0 3759 y Fs(b)-5 b(e)35 b(the)f(symmetric)h(algebr)-5 b(a)34 b(over)g Ft(V)22 b Fs(.)45 b(Then)34 b(we)g(have)149 3904 y Fx(1.)42 b Ft(\023)47 b Fx(:)g Ft(V)68 b Fu(\000)-58 b(!)48 b Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))45 b Fs(is)g(inje)-5 b(ctive)45 b(\(we)f(wil)5 b(l)45 b(identify)g(the)g(elements)f Ft(\023)p Fx(\()p Ft(v)t Fx(\))i Fs(and)e Ft(v)49 b Fs(for)c(al)5 b(l)267 4021 y Ft(v)31 b Fu(2)d Ft(V)22 b Fs(\),)149 4137 y Fx(2.)42 b Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))28 b(=)f Fu(f)668 4062 y Ff(P)773 4166 y Fq(n;)830 4151 y Fn(\026)836 4166 y Fq(i)880 4137 y Ft(v)927 4152 y Fq(i)951 4161 y Fc(1)1013 4137 y Fu(\001)21 b Ft(:)c(:)g(:)22 b Fu(\001)g Ft(v)1296 4152 y Fq(i)1320 4160 y Fk(n)1367 4137 y Fu(j)1387 4115 y Fx(\026)1395 4137 y Ft(i)27 b Fx(=)h(\()p Ft(i)1630 4152 y Fn(1)1669 4137 y Ft(;)17 b(:)g(:)g(:)33 b(;)17 b(i)1938 4152 y Fq(n)1985 4137 y Fx(\))35 b Fs(multiindex)f(of)g(length)h Ft(n)p Fu(g)p Ft(;)149 4262 y Fx(3.)42 b Fs(if)48 b Ft(f)65 b Fx(:)53 b Ft(V)76 b Fu(\000)-58 b(!)54 b Ft(A)49 b Fs(is)f(a)h(homomorphism)e (of)h Fp(K)18 b Fs(-mo)-5 b(dules)54 b(satisfying)48 b Ft(f)11 b Fx(\()p Ft(v)t Fx(\))32 b Fu(\001)g Ft(f)11 b Fx(\()p Ft(v)3410 4226 y Fr(0)3433 4262 y Fx(\))53 b(=)267 4378 y Ft(f)11 b Fx(\()p Ft(v)415 4342 y Fr(0)438 4378 y Fx(\))h Fu(\001)g Ft(f)f Fx(\()p Ft(v)t Fx(\))29 b Fs(for)h(al)5 b(l)30 b Ft(v)t(;)17 b(v)1175 4342 y Fr(0)1225 4378 y Fu(2)28 b Ft(V)21 b Fs(,)31 b Ft(A)g 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Fx(\))39 b(is)0 5099 y(the)c(comm)m(utativ)m (e)f(algebra)g(de\014ned)i(in)f(8.5.3.)50 b(to)34 b(construct)i Ft(g)f Fx(:)d Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))32 b Fu(\000)-60 b(!)32 b Ft(D)s Fx(\()p Ft(V)21 b Fx(\))35 b(suc)m(h)h(that)0 5216 y Ft(g)25 b Fu(\016)d Ft(\023)29 b Fx(=)e Ft(j)6 b Fx(.)43 b(Since)33 b Ft(j)39 b Fx(is)32 b(injectiv)m(e)g(so)h(is)f Ft(\023)p Fx(.)p eop %%Page: 22 22 22 21 bop 0 236 a Fo(22)1456 b(8.)77 b(TOOLBO)n(X)149 425 y Fx(2.)43 b(Let)31 b Ft(B)i Fx(:=)27 b Fu(f)728 350 y Ff(P)833 454 y Fq(n;)890 439 y Fn(\026)896 454 y Fq(i)940 425 y Ft(v)987 440 y Fq(i)1011 449 y Fc(1)1069 425 y Fu(\001)17 b Ft(:)g(:)g(:)h Fu(\001)g Ft(v)1340 440 y Fq(i)1364 448 y Fk(n)1411 425 y Fu(j)1431 403 y Fx(\026)1439 425 y Ft(i)27 b Fx(=)h(\()p Ft(i)1674 440 y Fn(1)1713 425 y Ft(;)17 b(:)g(:)g(:)33 b(;)17 b(i)1982 440 y Fq(n)2029 425 y Fx(\))32 b(m)m(ultiindex)f(of)h(length)g Ft(n)p Fu(g)p Fx(.)43 b(Ob)m(viously)0 550 y Ft(B)d Fx(is)35 b(the)g(subalgebra)g(of)f Ft(S)6 b Fx(\()p Ft(V)22 b Fx(\))34 b(generated)i(b)m(y)g(the)g(elemen)m(ts)f(of)g Ft(V)21 b Fx(.)51 b(Let)35 b Ft(j)j Fx(:)32 b Ft(B)37 b Fu(\000)-60 b(!)32 b Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))35 b(b)s(e)0 666 y(the)k(em)m(b)s(edding)g(homomorphism.)60 b(Then)40 b Ft(\023)f Fx(:)g Ft(V)60 b Fu(\000)-60 b(!)39 b Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))39 b(factors)g(through)g(a)f(linear) g(map)0 783 y Ft(\023)34 746 y Fr(0)85 783 y Fx(:)28 b Ft(V)49 b Fu(\000)-60 b(!)28 b Ft(B)5 b Fx(.)44 b(In)33 b(the)g(follo)m(wing)d(diagram)1226 990 y Ft(V)430 b(B)p 1333 958 351 4 v 1601 956 a Fl(-)1480 937 y Ft(\023)1514 901 y Fr(0)2130 981 y Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))p 1821 958 281 4 v 2019 956 a Fl(-)1938 918 y Ft(j)1713 1469 y(B)343 b(S)6 b Fx(\()p Ft(V)21 b Fx(\))p 1821 1446 V 2019 1444 a Fl(-)1938 1405 y Ft(j)1391 1237 y(\023)1425 1201 y Fr(0)1333 1108 y Fl(@)1417 1191 y(@)1500 1274 y(@)1583 1357 y(@)1601 1376 y(@)-83 b(R)p 1751 1376 4 351 v 69 w(?)1572 1227 y Fx(id)1653 1242 y Fq(B)p 2239 1376 V 2240 1376 a Fl(?)2279 1223 y Ft(j)6 b(p)2057 1212 y(p)2089 1108 y Fl(\000)2006 1191 y(\000)1923 1274 y(\000)1840 1357 y(\000)1821 1376 y(\000)-83 b(\011)0 1618 y Fx(w)m(e)38 b(ha)m(v)m(e)g(id)458 1633 y Fq(B)535 1618 y Fu(\016)p Ft(\023)619 1582 y Fr(0)678 1618 y Fx(=)d Ft(\023)823 1582 y Fr(0)846 1618 y Fx(,)j Ft(p)f Fx(with)f Ft(p)25 b Fu(\016)g Ft(j)31 b Fu(\016)25 b Ft(\023)1552 1582 y Fr(0)1611 1618 y Fx(=)34 b Ft(p)25 b Fu(\016)g Ft(\023)35 b Fx(=)g Ft(\023)2084 1582 y Fr(0)2145 1618 y Fx(exists)i(since)g Ft(\023)2695 1582 y Fr(0)2756 1618 y Fx(is)f(a)h(homomorphism)0 1734 y(of)47 b Fp(K)18 b Fx(-mo)s(dules)52 b(satisfying)47 b Ft(\023)1118 1698 y Fr(0)1141 1734 y Fx(\()p Ft(v)t Fx(\))32 b Fu(\001)g Ft(\023)1394 1698 y Fr(0)1418 1734 y Fx(\()p Ft(v)1507 1698 y Fr(0)1530 1734 y Fx(\))53 b(=)h Ft(\023)1785 1698 y Fr(0)1808 1734 y Fx(\()p Ft(v)1897 1698 y Fr(0)1920 1734 y Fx(\))33 b Fu(\001)f Ft(\023)2085 1698 y Fr(0)2109 1734 y Fx(\()p Ft(v)t Fx(\))47 b(for)g(all)e Ft(v)t(;)17 b(v)2743 1698 y Fr(0)2819 1734 y Fu(2)54 b Ft(V)22 b Fx(.)88 b(Because)49 b(of)0 1850 y Ft(j)6 b(p)23 b Fu(\016)f Ft(\023)30 b Fx(=)f Ft(j)g Fu(\016)23 b Ft(\023)535 1814 y Fr(0)588 1850 y Fx(=)29 b Ft(\023)h Fx(=)f(id)943 1866 y Fq(S)t Fn(\()p Fq(V)16 b Fn(\))1122 1850 y Fu(\016)p Ft(\023)34 b Fx(w)m(e)h(get)e Ft(j)6 b(p)29 b Fx(=)h(id)1859 1866 y Fq(S)t Fn(\()p Fq(V)16 b Fn(\))2021 1850 y Fx(,)34 b(hence)h(the)f(em)m(b)s(edding)f Ft(j)40 b Fx(is)33 b(surjectiv)m(e)0 1966 y(and)g(th)m(us)g(the)g(iden) m(tit)m(y)-8 b(.)149 2083 y(3.)44 b(is)32 b(precisely)h(the)g (de\014nition)e(of)h(the)h(induced)g(homomorphism.)p 3529 2083 4 66 v 3533 2020 59 4 v 3533 2083 V 3591 2083 4 66 v 149 2258 a Fv(Prop)s(osition)j(8.5.12.)49 b Fs(L)-5 b(et)38 b Ft(V)59 b Fs(b)-5 b(e)37 b(a)g Fp(K)17 b Fs(-mo)-5 b(dule.)58 b(The)36 b(symmetric)h(algebr)-5 b(a)37 b Fx(\()p Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))p Ft(;)c(\023)p Fx(\))37 b Fs(is)0 2374 y(c)-5 b(ommutative)34 b(and)h(satis\014es)f (the)h(fol)5 b(lowing)33 b(universal)h(pr)-5 b(op)g(erty:)149 2491 y(for)50 b(e)-5 b(ach)48 b(c)-5 b(ommutative)49 b Fp(K)17 b Fs(-algebr)-5 b(a)55 b Ft(A)49 b Fs(and)g(for)g(e)-5 b(ach)48 b(homomorphism)f(of)i Fp(K)18 b Fs(-mo)-5 b(dules)0 2607 y Ft(f)45 b Fx(:)35 b Ft(V)56 b Fu(\000)-58 b(!)34 b Ft(A)39 b Fs(ther)-5 b(e)39 b(exists)f(a)g(unique)g(homomorphism)f (of)h Fp(K)17 b Fs(-algebr)-5 b(as)44 b Ft(g)38 b Fx(:)c Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))35 b Fu(\000)-58 b(!)35 b Ft(A)j Fs(such)0 2723 y(that)d(the)g(diagr)-5 b(am)1482 2838 y Ft(V)360 b(S)6 b Fx(\()p Ft(V)21 b Fx(\))p 1589 2815 281 4 v 1787 2813 a Fl(-)1713 2795 y Ft(\023)1646 3082 y(f)1589 2964 y Fl(@)1672 3047 y(@)1755 3131 y(@)1838 3214 y(@)1857 3232 y(@)-83 b(R)1972 3335 y Ft(A)p 2007 3232 4 351 v 2009 3232 a Fl(?)2048 3069 y Ft(g)0 3439 y Fs(c)-5 b(ommutes.)149 3615 y Fi(Pr)n(oof.)42 b Fx(Comm)m(utativit)m (y)30 b(follo)m(ws)g(from)g(the)i(comm)m(utativit)m(y)e(of)h(the)g (generators:)44 b Ft(v)t(v)3474 3579 y Fr(0)3524 3615 y Fx(=)0 3731 y Ft(v)51 3695 y Fr(0)74 3731 y Ft(v)e Fx(whic)m(h)d(carries)g(o)m(v)m(er)h(to)e(the)h(elemen)m(ts)g(of)f(the) h(form)2210 3656 y Ff(P)2315 3760 y Fq(n;)2372 3745 y Fn(\026)2378 3760 y Fq(i)2423 3731 y Ft(v)2470 3746 y Fq(i)2494 3755 y Fc(1)2559 3731 y Fu(\001)26 b Ft(:)17 b(:)g(:)25 b Fu(\001)h Ft(v)2854 3746 y Fq(i)2878 3754 y Fk(n)2925 3731 y Fx(.)62 b(The)39 b(univ)m(ersal)0 3856 y(prop)s(ert)m(y)29 b(follo)m(ws)e(since)i(the)g(de\014ning)f (condition)f Ft(f)11 b Fx(\()p Ft(v)t Fx(\))j Fu(\001)g Ft(f)d Fx(\()p Ft(v)2287 3820 y Fr(0)2308 3856 y Fx(\))28 b(=)g Ft(f)11 b Fx(\()p Ft(v)2626 3820 y Fr(0)2648 3856 y Fx(\))j Fu(\001)g Ft(f)d Fx(\()p Ft(v)t Fx(\))27 b(for)h(all)e Ft(v)t(;)17 b(v)3377 3820 y Fr(0)3427 3856 y Fu(2)28 b Ft(V)0 3972 y Fx(is)k(automatically)d(satis\014ed.)p 3529 3972 4 66 v 3533 3910 59 4 v 3533 3972 V 3591 3972 4 66 v 149 4148 a Fv(Prop)s(osition)36 b(8.5.13.)49 b Fs(Given)34 b(a)f Fp(K)18 b Fs(-mo)-5 b(dule)39 b Ft(V)21 b Fs(.)45 b(Then)33 b(ther)-5 b(e)33 b(exists)h(a)f(symmetric)h(alge-)0 4264 y(br)-5 b(a)35 b Fx(\()p Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))p Ft(;)c(\023)p Fx(\))p Fs(.)149 4440 y Fi(Pr)n(oof.)42 b Fx(De\014ne)35 b Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))29 b(:=)h Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))p Ft(=I)41 b Fx(where)35 b Ft(I)i Fx(=)29 b Fu(h)p Ft(v)t(v)2187 4403 y Fr(0)2233 4440 y Fu(\000)23 b Ft(v)2384 4403 y Fr(0)2407 4440 y Ft(v)t Fu(j)p Ft(v)t(;)17 b(v)2632 4403 y Fr(0)2684 4440 y Fu(2)29 b Ft(V)22 b Fu(i)33 b Fx(is)g(the)h(t)m(w)m(o-sided)0 4556 y(ideal)29 b(generated)j(b)m(y)f(the)g(elemen)m(ts)g Ft(v)t(v)1472 4520 y Fr(0)1513 4556 y Fu(\000)19 b Ft(v)1660 4520 y Fr(0)1683 4556 y Ft(v)t Fx(.)42 b(Let)31 b Ft(\023)g Fx(b)s(e)g(the)g(canonical)e(map)h Ft(V)49 b Fu(\000)-60 b(!)28 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))28 b Fu(\000)-60 b(!)0 4672 y Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\).)50 b(Then)36 b(the)f(univ)m(ersal)g(prop)s(ert)m(y)g(is)g(easily)f(v)m(eri\014ed)h (b)m(y)h(the)f(homomorphism)d(theorem)0 4788 y(for)g(algebras.)p 3529 4788 V 3533 4726 59 4 v 3533 4788 V 3591 4788 4 66 v 149 4964 a Fv(Prop)s(osition)k(8.5.14.)49 b Fs(The)54 b(c)-5 b(onstruction)54 b(of)g(symmetric)g(algebr)-5 b(as)53 b Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))54 b Fs(de\014nes)f(a)0 5080 y(functor)45 b Ft(S)52 b Fx(:)47 b Fp(K)18 b Fs(-)5 b Fv(Mo)s(d)48 b Fu(\000)-58 b(!)47 b Fp(K)17 b Fs(-)6 b Fv(cAlg)44 b Fs(that)h(is)g(left)g(adjoint)g(to)g(the)g(underlying)g (functor)g Ft(U)57 b Fx(:)0 5196 y Fp(K)17 b Fs(-)6 b Fv(cAlg)27 b Fu(\000)-58 b(!)28 b Fp(K)17 b Fs(-)6 b Fv(Mo)s(d)q Fs(.)p eop %%Page: 23 23 23 22 bop 1530 236 a Fo(5.)77 b(ALGEBRAS)1427 b(23)149 425 y Fi(Pr)n(oof.)42 b Fx(F)-8 b(ollo)m(ws)31 b(from)h(the)h(univ)m (ersal)f(prop)s(ert)m(y)i(and)e(8.9.16.)p 3529 425 4 66 v 3533 363 59 4 v 3533 425 V 3591 425 4 66 v 149 602 a Fv(Problem)k(8.5.7.)49 b Fx(Let)42 b Ft(X)49 b Fx(b)s(e)41 b(a)g(set)h(and)f Ft(V)64 b Fx(:=)42 b Fp(K)18 b Ft(X)55 b Fx(b)s(e)41 b(the)h(free)f Fp(K)18 b Fx(-mo)s(dule)45 b(o)m(v)m(er)d Ft(X)8 b Fx(.)0 718 y(Sho)m(w)32 b(that)f Ft(X)k Fu(\000)-60 b(!)28 b Ft(V)49 b Fu(\000)-60 b(!)28 b Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))31 b(de\014nes)h(a)f Fs(fr)-5 b(e)g(e)33 b(c)-5 b(ommutative)33 b(algebr)-5 b(a)30 b Fx(o)m(v)m(er)i Ft(X)8 b Fx(,)31 b(i.e.)43 b(for)30 b(ev)m(ery)0 835 y(comm)m(utativ)m(e)21 b Fp(K)d Fx(-algebra)26 b Ft(A)d Fx(and)f(ev)m(ery)i(map)e Ft(f)38 b Fx(:)28 b Ft(X)35 b Fu(\000)-60 b(!)28 b Ft(A)22 b Fx(there)h(is)f(a)g(unique)h (homomorphism)0 951 y(of)32 b Fp(K)18 b Fx(-algebras)37 b Ft(g)31 b Fx(:)d Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))28 b Fu(\000)-60 b(!)27 b Ft(A)33 b Fx(suc)m(h)h(that)e(the)h(diagram)1479 1104 y Ft(X)341 b(S)6 b Fx(\()p Ft(V)21 b Fx(\))p 1597 1081 276 4 v 1790 1079 a Fl(-)1648 1348 y Ft(f)1592 1230 y Fl(@)1675 1313 y(@)1758 1396 y(@)1841 1479 y(@)1860 1498 y(@)-83 b(R)1975 1601 y Ft(A)p 2009 1498 4 351 v 2011 1498 a Fl(?)2050 1335 y Ft(g)0 1728 y Fx(comm)m(utes.)149 1844 y(The)40 b(algebra)e Fp(K)18 b Fx([)p Ft(X)8 b Fx(])45 b(:=)38 b Ft(S)6 b Fx(\()p Fp(K)18 b Ft(X)8 b Fx(\))44 b(is)39 b(called)f(the)i Fs(p)-5 b(olynomial)39 b(ring)i(over)f Fp(K)64 b Fs(in)41 b(the)g(\(c)-5 b(om-)0 1960 y(muting\))35 b(variables)e Ft(X)8 b Fx(.)149 2077 y(2.)64 b(Let)39 b Ft(S)6 b Fx(\()p Ft(V)22 b Fx(\))39 b(and)g Ft(\023)h Fx(:)f Ft(V)61 b Fu(\000)-60 b(!)39 b Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))39 b(b)s(e)h(a)f(symmetric)g(algebra.)62 b(Sho)m(w)40 b(that)f(there)h(is)f(a)0 2193 y(unique)c(homomorphism)c (\001)g(:)f Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))31 b Fu(\000)-60 b(!)30 b Ft(S)6 b Fx(\()p Ft(V)22 b Fx(\))h Fu(\012)g Ft(S)6 b Fx(\()p Ft(V)22 b Fx(\))34 b(with)g(\001\()p Ft(v)t Fx(\))c(=)g Ft(v)d Fu(\012)d Fx(1)f(+)g(1)g Fu(\012)h Ft(v)37 b Fx(for)d(all)0 2309 y Ft(v)d Fu(2)d Ft(V)22 b Fx(.)149 2425 y(3.)44 b(Sho)m(w)33 b(that)f(\(\001)23 b Fu(\012)f Fx(1\)\001)28 b(=)g(\(1)22 b Fu(\012)g Fx(\001\)\001)28 b(:)g Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))28 b Fu(\000)-60 b(!)28 b Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))h Fu(\012)h Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))h Fu(\012)h Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\).)149 2542 y(4.)61 b(Sho)m(w)40 b(that)e(there)h(is)f(a)g(unique)h(homomorphism)d(of)i(algebras)g Ft(")f Fx(:)h Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))38 b Fu(\000)-60 b(!)37 b Fp(K)62 b Fx(with)0 2658 y Ft(")p Fx(\()p Ft(v)t Fx(\))27 b(=)g(0)33 b(for)f(all)e Ft(v)i Fu(2)c Ft(V)21 b Fx(.)149 2774 y(5.)44 b(Sho)m(w)33 b(that)f(\()p Ft(")22 b Fu(\012)h Fx(1\)\001)k(=)h(\(1)22 b Fu(\012)g Ft(")p Fx(\)\001)28 b(=)g(id)1828 2790 y Fq(S)t Fn(\()p Fq(V)16 b Fn(\))1990 2774 y Fx(.)149 2890 y(6.)42 b(Sho)m(w)29 b(that)f(there)h(is)e(a)h(unique)h(homomorphism)c(of)j(algebras)f Ft(S)34 b Fx(:)27 b Ft(S)6 b Fx(\()p Ft(V)22 b Fx(\))27 b Fu(\000)-60 b(!)28 b Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))28 b(with)0 3007 y Ft(S)6 b Fx(\()p Ft(v)t Fx(\))27 b(=)h Fu(\000)p Ft(v)t Fx(.)149 3123 y(7.)44 b(Sho)m(w)33 b(that)f(the)h (diagrams)1300 3281 y Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))241 b Fp(K)p 1548 3258 185 4 v 1650 3256 a Fl(-)1624 3237 y Fq(")2080 3281 y Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))p 1867 3258 V 1969 3256 a Fl(-)1940 3224 y Fq(\021)1129 3671 y Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))h Fu(\012)h Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))218 b Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))h Fu(\012)h Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))p 1720 3648 160 4 v 1797 3646 a Fl(-)1732 3620 y Fn(1)p Fr(\012)p Fq(S)1732 3710 y(S)t Fr(\012)p Fn(1)p 1408 3578 4 254 v 1410 3578 a Fl(?)1312 3474 y Fn(\001)p 2188 3578 V 2190 3407 a Fl(6)2229 3474 y Fr(r)0 3831 y Fx(comm)m(ute.)149 4008 y Fv(De\014nition)37 b(8.5.15.)49 b Fx(Let)44 b Fp(K)68 b Fx(b)s(e)45 b(a)f(comm)m(utativ)m(e)f(ring.)78 b(Let)44 b Ft(V)66 b Fx(b)s(e)45 b(a)f Fp(K)17 b Fx(-mo)s(dule.)83 b(A)0 4124 y Fp(K)17 b Fx(-algebra)41 b Ft(E)6 b Fx(\()p Ft(V)21 b Fx(\))36 b(together)h(with)e(a)h(homomorphism)d(of)j Fp(K)17 b Fx(-mo)s(dules)41 b Ft(\023)34 b Fx(:)f Ft(V)56 b Fu(\000)-60 b(!)33 b Ft(E)6 b Fx(\()p Ft(V)22 b Fx(\),)36 b(suc)m(h)0 4241 y(that)e Ft(\023)p Fx(\()p Ft(v)t Fx(\))374 4204 y Fn(2)443 4241 y Fx(=)c(0)j(for)h(all)d Ft(v)j Fu(2)c Ft(V)22 b Fx(,)34 b(is)g(called)e(an)i Fs(exterior)i(algebr)-5 b(a)35 b(or)h(Gr)-5 b(assmann)35 b(algebr)-5 b(a)35 b(over)0 4357 y Ft(V)61 b Fx(if)38 b(for)h(eac)m(h)i Fp(K)17 b Fx(-algebra)44 b Ft(A)c Fx(and)f(for)g(eac)m(h)i(homomorphism)c(of)i Fp(K)17 b Fx(-mo)s(dules)44 b Ft(f)51 b Fx(:)39 b Ft(V)61 b Fu(\000)-60 b(!)40 b Ft(A)p Fx(,)0 4473 y(suc)m(h)31 b(that)e Ft(f)11 b Fx(\()p Ft(v)t Fx(\))611 4437 y Fn(2)677 4473 y Fx(=)28 b(0)h(for)g(all)e Ft(v)k Fu(2)e Ft(V)21 b Fx(,)30 b(there)g(exists)g(a)f(unique)h(homomorphism)d(of)i Fp(K)17 b Fx(-algebras)0 4589 y Ft(g)31 b Fx(:)d Ft(E)6 b Fx(\()p Ft(V)21 b Fx(\))28 b Fu(\000)-60 b(!)28 b Ft(A)k Fx(suc)m(h)i(that)f(the)g(diagram)1478 4753 y Ft(V)354 b(E)6 b Fx(\()p Ft(V)22 b Fx(\))p 1586 4730 274 4 v 1777 4728 a Fl(-)1705 4710 y Ft(\023)1642 4997 y(f)1586 4879 y Fl(@)1669 4962 y(@)1752 5045 y(@)1835 5128 y(@)1854 5147 y(@)-83 b(R)1969 5250 y Ft(A)p 2004 5147 4 351 v 2005 5147 a Fl(?)2044 4984 y Ft(g)p eop %%Page: 24 24 24 23 bop 0 236 a Fo(24)1456 b(8.)77 b(TOOLBO)n(X)0 425 y Fx(comm)m(utes.)149 541 y(The)34 b(m)m(ultiplication)28 b(in)k Ft(E)6 b Fx(\()p Ft(V)21 b Fx(\))33 b(is)f(usually)g(denoted)h (b)m(y)h Ft(u)21 b Fu(^)i Ft(v)t Fx(.)149 658 y Fs(Note:)61 b Fx(If)41 b(y)m(ou)g(w)m(an)m(t)h(to)f(de\014ne)h(a)f(homomorphism)d Ft(g)45 b Fx(:)d Ft(E)6 b Fx(\()p Ft(V)22 b Fx(\))42 b Fu(\000)-60 b(!)42 b Ft(A)f Fx(with)f(an)h(exterior)0 774 y(algebra)j(as)i(domain)d(y)m(ou)j(should)f(de\014ne)i(it)d(b)m(y)i (giving)e(a)h(homomorphism)d(of)j Fp(K)17 b Fx(-mo)s(dules)0 890 y(de\014ned)34 b(on)e Ft(V)55 b Fx(satisfying)31 b Ft(f)11 b Fx(\()p Ft(v)t Fx(\))1203 854 y Fn(2)1270 890 y Fx(=)27 b(0)32 b(for)h(all)d Ft(v)t(;)17 b(v)1885 854 y Fr(0)1935 890 y Fu(2)28 b Ft(V)22 b Fx(.)149 1099 y Fv(Problem)36 b(8.5.8.)49 b Fx(1.)k(Let)36 b Ft(f)44 b Fx(:)33 b Ft(V)54 b Fu(\000)-60 b(!)33 b Ft(A)j Fx(b)s(e)g(a)f (linear)f(map)h(satisfying)g Ft(f)11 b Fx(\()p Ft(v)t Fx(\))3080 1062 y Fn(2)3152 1099 y Fx(=)32 b(0)k(for)f(all)0 1215 y Ft(v)c Fu(2)d Ft(V)22 b Fx(.)43 b(Then)34 b Ft(f)11 b Fx(\()p Ft(v)t Fx(\))p Ft(f)g Fx(\()p Ft(v)910 1179 y Fr(0)932 1215 y Fx(\))28 b(=)f Fu(\000)p Ft(f)11 b Fx(\()p Ft(v)1326 1179 y Fr(0)1350 1215 y Fx(\))p Ft(f)g Fx(\()p Ft(v)t Fx(\))31 b(for)h(all)f Ft(v)t(;)17 b(v)2036 1179 y Fr(0)2086 1215 y Fu(2)28 b Ft(V)22 b Fx(.)149 1331 y(2.)48 b(Let)35 b(2)f(b)s(e)g(in)m(v)m(ertible)f(in)h Fp(K)57 b Fx(\(e.g.)49 b Fp(K)57 b Fx(a)34 b(\014eld)g(of)g(c)m (haracteristic)g Fu(6)p Fx(=)c(2\).)48 b(Let)34 b Ft(f)41 b Fx(:)31 b Ft(V)51 b Fu(\000)-60 b(!)31 b Ft(A)0 1447 y Fx(b)s(e)k(a)g(linear)e(map)h(satisfying)g Ft(f)11 b Fx(\()p Ft(v)t Fx(\))p Ft(f)g Fx(\()p Ft(v)1482 1411 y Fr(0)1504 1447 y Fx(\))32 b(=)f Fu(\000)p Ft(f)11 b Fx(\()p Ft(v)1906 1411 y Fr(0)1929 1447 y Fx(\))p Ft(f)g Fx(\()p Ft(v)t Fx(\))34 b(for)h(all)d Ft(v)t(;)17 b(v)2622 1411 y Fr(0)2676 1447 y Fu(2)32 b Ft(V)22 b Fx(.)50 b(Then)36 b Ft(f)11 b Fx(\()p Ft(v)t Fx(\))3373 1411 y Fn(2)3444 1447 y Fx(=)31 b(0)0 1564 y(for)h(all)f Ft(v)g Fu(2)d Ft(V)22 b Fx(.)149 1772 y Fv(Lemma)38 b(8.5.16.)49 b Fs(A)n(n)c(exterior)h(algebr)-5 b(a)45 b Fx(\()p Ft(E)6 b Fx(\()p Ft(V)22 b Fx(\))p Ft(;)17 b(\023)p Fx(\))46 b Fs(de\014ne)-5 b(d)44 b(by)i Ft(V)68 b Fs(is)45 b(unique)h(up)g(to)g (a)0 1888 y(unique)35 b(isomorphism.)149 2097 y Fi(Pr)n(oof.)42 b Fx(Let)33 b(\()p Ft(E)6 b Fx(\()p Ft(V)22 b Fx(\))p Ft(;)17 b(\023)p Fx(\))32 b(and)h(\()p Ft(E)1434 2061 y Fr(0)1457 2097 y Fx(\()p Ft(V)22 b Fx(\))p Ft(;)17 b(\023)1690 2061 y Fr(0)1713 2097 y Fx(\))33 b(b)s(e)f(exterior)h (algebras)f(o)m(v)m(er)h Ft(V)22 b Fx(.)43 b(Then)1755 2298 y Ft(V)1210 2529 y(\023)1575 2373 y Fl(\010)1492 2415 y(\010)1409 2457 y(\010)1326 2498 y(\010)1243 2540 y(\010)1160 2581 y(\010)1077 2623 y(\010)993 2664 y(\010)956 2683 y(\010)-83 b(\031)1611 2544 y Ft(\023)1645 2508 y Fr(0)1643 2415 y Fl(\000)1560 2498 y(\000)1477 2581 y(\000)1394 2664 y(\000)1375 2683 y(\000)g(\011)1943 2529 y Ft(\023)1862 2415 y Fl(@)1945 2498 y(@)2028 2581 y(@)2111 2664 y(@)2130 2683 y(@)g(R)2344 2544 y Ft(\023)2378 2508 y Fr(0)1931 2373 y Fl(H)2014 2415 y(H)2097 2457 y(H)2180 2498 y(H)2263 2540 y(H)2346 2581 y(H)2429 2623 y(H)2512 2664 y(H)2550 2683 y(H)g(j)703 2776 y Ft(E)6 b Fx(\()p Ft(V)22 b Fx(\))243 b Ft(E)1257 2740 y Fr(0)1280 2776 y Fx(\()p Ft(V)22 b Fx(\))p 964 2753 186 4 v 1067 2751 a Fl(-)1029 2732 y Ft(h)p 1464 2753 673 4 v 2054 2751 a Fl(-)1773 2732 y Ft(k)2165 2776 y(E)6 b Fx(\()p Ft(V)22 b Fx(\))243 b Ft(E)2719 2740 y Fr(0)2743 2776 y Fx(\()p Ft(V)21 b Fx(\))p 2427 2753 186 4 v 2530 2751 a Fl(-)2491 2732 y Ft(h)0 2989 y Fx(implies)30 b Ft(k)h Fx(=)c Ft(h)572 2953 y Fr(\000)p Fn(1)667 2989 y Fx(.)p 3529 2989 4 66 v 3533 2926 59 4 v 3533 2989 V 3591 2989 4 66 v 149 3197 a Fv(Prop)s(osition)36 b(8.5.17.)49 b Fx(\(Rules)26 b(of)f(computation)g(in)g(an)g(exterior)h(algebra\))h Fs(L)-5 b(et)29 b Fx(\()p Ft(E)6 b Fx(\()p Ft(V)22 b Fx(\))p Ft(;)17 b(\023)p Fx(\))0 3313 y Fs(b)-5 b(e)35 b(the)f(exterior)h(algebr)-5 b(a)34 b(over)h Ft(V)21 b Fs(.)45 b(Then)34 b(we)g(have)149 3480 y Fx(1.)42 b Ft(\023)k Fx(:)g Ft(V)67 b Fu(\000)-58 b(!)46 b Ft(E)6 b Fx(\()p Ft(V)22 b Fx(\))44 b Fs(is)h(inje)-5 b(ctive)43 b(\(we)h(wil)5 b(l)44 b(identify)h(the)f(elements)g Ft(\023)p Fx(\()p Ft(v)t Fx(\))h Fs(and)f Ft(v)k Fs(for)c(al)5 b(l)267 3596 y Ft(v)31 b Fu(2)d Ft(V)22 b Fs(\),)149 3712 y Fx(2.)42 b Ft(E)6 b Fx(\()p Ft(V)21 b Fx(\))28 b(=)g Fu(f)681 3638 y Ff(P)785 3741 y Fq(n;)842 3726 y Fn(\026)848 3741 y Fq(i)893 3712 y Ft(v)940 3727 y Fq(i)964 3736 y Fc(1)1025 3712 y Fu(^)22 b Ft(:)17 b(:)g(:)22 b Fu(^)h Ft(v)1386 3727 y Fq(i)1410 3735 y Fk(n)1456 3712 y Fu(j)1476 3690 y Fx(\026)1484 3712 y Ft(i)28 b Fx(=)g(\()p Ft(i)1720 3727 y Fn(1)1759 3712 y Ft(;)17 b(:)g(:)g(:)33 b(;)17 b(i)2028 3727 y Fq(n)2075 3712 y Fx(\))34 b Fs(multiindex)h(of)f(length)h Ft(n)p Fu(g)p Ft(;)149 3837 y Fx(3.)42 b Fs(if)48 b Ft(f)65 b Fx(:)53 b Ft(V)76 b Fu(\000)-58 b(!)54 b Ft(A)49 b Fs(is)f(a)h(homomorphism)e (of)h Fp(K)18 b Fs(-mo)-5 b(dules)54 b(satisfying)48 b Ft(f)11 b Fx(\()p Ft(v)t Fx(\))32 b Fu(\001)g Ft(f)11 b Fx(\()p Ft(v)3410 3801 y Fr(0)3433 3837 y Fx(\))53 b(=)267 3954 y Fu(\000)p Ft(f)11 b Fx(\()p Ft(v)492 3917 y Fr(0)515 3954 y Fx(\))28 b Fu(\001)f Ft(f)11 b Fx(\()p Ft(v)t Fx(\))42 b Fs(for)h(al)5 b(l)42 b Ft(v)t(;)17 b(v)1321 3917 y Fr(0)1386 3954 y Fu(2)42 b Ft(V)22 b Fs(,)44 b Ft(A)f Fs(is)f(a)h Fp(K)17 b Fs(-algebr)-5 b(a,)50 b(and)42 b Ft(g)j Fx(:)e Ft(E)6 b Fx(\()p Ft(V)21 b Fx(\))42 b Fu(\000)-58 b(!)43 b Ft(A)g Fs(is)f(the)267 4070 y(induc)-5 b(e)g(d)34 b(homomorphism)f Fp(K)17 b Fs(-algebr)-5 b(as,)40 b(then)838 4303 y Ft(g)t Fx(\()927 4208 y Ff(X)955 4427 y Fq(n;)1012 4412 y Fn(\026)1018 4427 y Fq(i)1086 4303 y Ft(v)1133 4318 y Fq(i)1157 4327 y Fc(1)1219 4303 y Fu(^)22 b Ft(:)17 b(:)g(:)22 b Fu(^)g Ft(v)1579 4318 y Fq(i)1603 4326 y Fk(n)1650 4303 y Fx(\))28 b(=)1819 4208 y Ff(X)1848 4427 y Fq(n;)1905 4412 y Fn(\026)1911 4427 y Fq(i)1980 4303 y Ft(f)11 b Fx(\()p Ft(v)2124 4318 y Fq(i)2148 4327 y Fc(1)2186 4303 y Fx(\))22 b Fu(\001)g Ft(:)17 b(:)g(:)22 b Fu(\001)g Ft(f)11 b Fx(\()p Ft(v)2627 4318 y Fq(i)2651 4326 y Fk(n)2697 4303 y Fx(\))p Ft(:)149 4626 y Fi(Pr)n(oof.)42 b Fx(1.)64 b(Use)40 b(the)f(em)m(b)s(edding)g (homomorphism)e Ft(j)45 b Fx(:)39 b Ft(V)60 b Fu(\000)-60 b(!)40 b Ft(D)s Fx(\()p Ft(V)21 b Fx(\),)41 b(where)f Ft(D)s Fx(\()p Ft(V)21 b Fx(\))39 b(is)0 4742 y(the)28 b(algebra)f(de\014ned)i(in)e(8.5.3.)41 b(to)28 b(construct)h Ft(g)h Fx(:)e Ft(E)6 b Fx(\()p Ft(V)22 b Fx(\))27 b Fu(\000)-60 b(!)28 b Ft(D)s Fx(\()p Ft(V)21 b Fx(\))28 b(suc)m(h)h(that)f Ft(g)16 b Fu(\016)c Ft(\023)27 b Fx(=)h Ft(j)6 b Fx(.)42 b(Since)0 4858 y Ft(j)c Fx(is)33 b(injectiv)m(e)f(so)h(is)f Ft(\023)p Fx(.)149 4974 y(2.)40 b(Let)23 b Ft(B)33 b Fx(:=)28 b Fu(f)718 4900 y Ff(P)823 5003 y Fq(n;)880 4988 y Fn(\026)886 5003 y Fq(i)930 4974 y Ft(v)977 4989 y Fq(i)1001 4998 y Fc(1)1042 4974 y Fu(^)r Ft(:)17 b(:)g(:)r Fu(^)r Ft(v)1342 4989 y Fq(i)1366 4997 y Fk(n)1413 4974 y Fu(j)1433 4952 y Fx(\026)1441 4974 y Ft(i)28 b Fx(=)f(\()p Ft(i)1676 4989 y Fn(1)1716 4974 y Ft(;)17 b(:)g(:)g(:)32 b(;)17 b(i)1984 4989 y Fq(n)2031 4974 y Fx(\))33 b(m)m(ultiindex)e(of)h (length)g Ft(n)p Fu(g)p Fx(.)40 b(Ob)m(viously)0 5099 y Ft(B)f Fx(is)33 b(the)i(subalgebra)e(of)h Ft(E)6 b Fx(\()p Ft(V)21 b Fx(\))34 b(generated)h(b)m(y)g(the)f(elemen)m(ts)g (of)g Ft(V)22 b Fx(.)47 b(Let)34 b Ft(j)i Fx(:)30 b Ft(B)35 b Fu(\000)-60 b(!)30 b Ft(E)6 b Fx(\()p Ft(V)22 b Fx(\))34 b(b)s(e)0 5216 y(the)39 b(em)m(b)s(edding)f(homomorphism.)58 b(Then)39 b Ft(\023)f Fx(:)f Ft(V)59 b Fu(\000)-60 b(!)38 b Ft(E)6 b Fx(\()p Ft(V)21 b Fx(\))38 b(factors)h(through)f(a)g(linear) f(map)p eop %%Page: 25 25 25 24 bop 1530 236 a Fo(5.)77 b(ALGEBRAS)1427 b(25)0 425 y Ft(\023)34 389 y Fr(0)85 425 y Fx(:)28 b Ft(V)49 b Fu(\000)-60 b(!)28 b Ft(B)5 b Fx(.)44 b(In)33 b(the)g(follo)m(wing)d (diagram)1226 631 y Ft(V)430 b(B)p 1333 599 351 4 v 1601 597 a Fl(-)1480 578 y Ft(\023)1514 542 y Fr(0)2124 622 y Ft(E)6 b Fx(\()p Ft(V)21 b Fx(\))p 1821 599 274 4 v 2012 597 a Fl(-)1934 558 y Ft(j)1713 1109 y(B)337 b(E)6 b Fx(\()p Ft(V)21 b Fx(\))p 1821 1086 V 2012 1084 a Fl(-)1934 1046 y Ft(j)1391 877 y(\023)1425 841 y Fr(0)1333 748 y Fl(@)1417 831 y(@)1500 914 y(@)1583 997 y(@)1601 1016 y(@)-83 b(R)p 1751 1016 4 351 v 69 w(?)1572 868 y Fx(id)1653 883 y Fq(B)p 2239 1016 V 2240 1016 a Fl(?)2279 864 y Ft(j)6 b(p)2057 852 y(p)2089 748 y Fl(\000)2006 831 y(\000)1923 914 y(\000)1840 997 y(\000)1821 1016 y(\000)-83 b(\011)0 1256 y Fx(w)m(e)38 b(ha)m(v)m(e)g(id)458 1271 y Fq(B)535 1256 y Fu(\016)p Ft(\023)619 1220 y Fr(0)678 1256 y Fx(=)d Ft(\023)823 1220 y Fr(0)846 1256 y Fx(,)j Ft(p)f Fx(with)f Ft(p)25 b Fu(\016)g Ft(j)31 b Fu(\016)25 b Ft(\023)1552 1220 y Fr(0)1611 1256 y Fx(=)34 b Ft(p)25 b Fu(\016)g Ft(\023)35 b Fx(=)g Ft(\023)2084 1220 y Fr(0)2145 1256 y Fx(exists)i(since)g Ft(\023)2695 1220 y Fr(0)2756 1256 y Fx(is)f(a)h(homomorphism)0 1372 y(of)43 b Fp(K)18 b Fx(-mo)s(dules)48 b(satisfying)43 b Ft(\023)1106 1336 y Fr(0)1129 1372 y Fx(\()p Ft(v)t Fx(\))30 b Fu(\001)f Ft(\023)1377 1336 y Fr(0)1401 1372 y Fx(\()p Ft(v)1490 1336 y Fr(0)1513 1372 y Fx(\))46 b(=)h Fu(\000)p Ft(\023)1831 1336 y Fr(0)1855 1372 y Fx(\()p Ft(v)1944 1336 y Fr(0)1967 1372 y Fx(\))30 b Fu(\001)f Ft(\023)2126 1336 y Fr(0)2150 1372 y Fx(\()p Ft(v)t Fx(\))43 b(for)g(all)f Ft(v)t(;)17 b(v)2773 1336 y Fr(0)2842 1372 y Fu(2)47 b Ft(V)21 b Fx(.)77 b(Because)45 b(of)0 1489 y Ft(j)6 b(p)22 b Fu(\016)g Ft(\023)29 b Fx(=)f Ft(j)g Fu(\016)22 b Ft(\023)530 1452 y Fr(0)582 1489 y Fx(=)28 b Ft(\023)h Fx(=)f(id)934 1504 y Fq(E)t Fn(\()p Fq(V)16 b Fn(\))1122 1489 y Fu(\016)p Ft(\023)33 b Fx(w)m(e)h(get)f Ft(j)6 b(p)28 b Fx(=)g(id)1854 1504 y Fq(E)t Fn(\()p Fq(V)16 b Fn(\))2026 1489 y Fx(,)33 b(hence)h(the)f(em)m(b)s(edding)g Ft(j)38 b Fx(is)33 b(surjectiv)m(e)0 1605 y(and)g(th)m(us)g Ft(j)39 b Fx(is)32 b(the)h(iden)m(tit)m(y)-8 b(.)149 1721 y(3.)44 b(is)32 b(precisely)h(the)g(de\014nition)e(of)h(the)h(induced)g(homomorphism.)p 3529 1721 4 66 v 3533 1659 59 4 v 3533 1721 V 3591 1721 4 66 v 149 1889 a Fv(Prop)s(osition)j(8.5.18.)49 b Fs(Given)29 b(a)g Fp(K)17 b Fs(-mo)-5 b(dule)35 b Ft(V)21 b Fs(.)43 b(Then)28 b(ther)-5 b(e)29 b(exists)g(an)f(exterior)h(algebr)-5 b(a)0 2006 y Fx(\()p Ft(E)6 b Fx(\()p Ft(V)21 b Fx(\))p Ft(;)c(\023)p Fx(\))p Fs(.)149 2174 y Fi(Pr)n(oof.)42 b Fx(De\014ne)i Ft(E)6 b Fx(\()p Ft(V)21 b Fx(\))45 b(:=)f Ft(T)14 b Fx(\()p Ft(V)22 b Fx(\))p Ft(=I)50 b Fx(where)43 b Ft(I)53 b Fx(=)44 b Fu(h)p Ft(v)2236 2138 y Fn(2)2275 2174 y Fu(j)p Ft(v)k Fu(2)e Ft(V)21 b Fu(i)42 b Fx(is)g(the)h(t)m(w)m (o-sided)g(ideal)0 2290 y(generated)g(b)m(y)h(the)f(elemen)m(ts)g Ft(v)1238 2254 y Fn(2)1277 2290 y Fx(.)73 b(Let)43 b Ft(\023)g Fx(b)s(e)f(the)h(canonical)e(map)h Ft(V)66 b Fu(\000)-60 b(!)45 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))45 b Fu(\000)-60 b(!)44 b Ft(E)6 b Fx(\()p Ft(V)22 b Fx(\).)0 2406 y(Then)49 b(the)g(univ)m(ersal)f(prop)s(ert)m(y)g(is)g(easily)f(v) m(eri\014ed)i(b)m(y)g(the)f(homomorphism)e(theorem)h(for)0 2523 y(algebras.)p 3529 2523 V 3533 2460 59 4 v 3533 2523 V 3591 2523 4 66 v 149 2691 a Fv(Problem)36 b(8.5.9.)49 b Fx(1.)60 b(Let)38 b Ft(V)59 b Fx(b)s(e)39 b(a)e(\014nite)h (dimensional)d(v)m(ector)40 b(space)f(of)e(dimension)g Ft(n)p Fx(.)0 2807 y(Sho)m(w)46 b(that)g Ft(E)6 b Fx(\()p Ft(V)21 b Fx(\))45 b(is)g(\014nite)h(dimensional)d(of)i(dimension)f(2) 2343 2771 y Fq(n)2389 2807 y Fx(.)83 b(\(Hin)m(t:)68 b(The)47 b(homogeneous)0 2961 y(comp)s(onen)m(ts)33 b Ft(E)615 2924 y Fq(i)643 2961 y Fx(\()p Ft(V)22 b Fx(\))32 b(ha)m(v)m(e)i(dimension)1516 2844 y Fb(\022)1619 2910 y Fa(n)1630 3010 y(i)1710 2844 y Fb(\023)1771 2961 y Fx(.)149 3123 y(2.)89 b(Sho)m(w)49 b(that)e(the)i(symmetric)e(group)g Ft(S)1838 3138 y Fq(n)1933 3123 y Fx(op)s(erates)h(\(from)e(the)j (left\))e(on)g Ft(T)3248 3086 y Fq(n)3295 3123 y Fx(\()p Ft(V)21 b Fx(\))48 b(b)m(y)0 3239 y Ft(\033)t Fx(\()p Ft(v)144 3254 y Fn(1)206 3239 y Fu(\012)22 b Ft(:)17 b(:)g(:)22 b Fu(\012)h Ft(v)589 3254 y Fq(n)636 3239 y Fx(\))k(=)h Ft(v)852 3255 y Fq(\033)894 3237 y Fh(\000)p Fc(1)977 3255 y Fn(\(1\))1094 3239 y Fu(\012)22 b Ft(:)17 b(:)g(:)22 b Fu(\012)h Ft(v)1477 3255 y Fq(\033)1519 3237 y Fh(\000)p Fc(1)1602 3255 y Fn(\()p Fq(n)p Fn(\))1736 3239 y Fx(with)33 b Ft(\033)e Fu(2)d Ft(S)2199 3254 y Fq(n)2279 3239 y Fx(and)33 b Ft(v)2516 3254 y Fq(i)2572 3239 y Fu(2)28 b Ft(V)21 b Fx(.)149 3355 y(3.)60 b(A)38 b(tensor)h Ft(a)e Fu(2)g Ft(T)957 3319 y Fq(n)1004 3355 y Fx(\()p Ft(V)21 b Fx(\))38 b(is)g(called)f Fs(a)i(symmetric)h(tensor) d Fx(if)g Ft(\033)t Fx(\()p Ft(a)p Fx(\))g(=)g Ft(a)h Fx(for)g(all)e Ft(\033)41 b Fu(2)c Ft(S)3526 3370 y Fq(n)3573 3355 y Fx(.)0 3483 y(Let)191 3457 y(^)175 3483 y Ft(S)241 3447 y Fq(n)288 3483 y Fx(\()p Ft(V)21 b Fx(\))32 b(b)s(e)h(the)g (subspace)i(of)d(symmetric)g(tensors)h(in)f Ft(T)2285 3447 y Fq(n)2332 3483 y Fx(\()p Ft(V)21 b Fx(\).)149 3599 y(a\))33 b(Sho)m(w)g(that)f Fu(S)k Fx(:)28 b Ft(T)959 3563 y Fq(n)1005 3599 y Fx(\()p Ft(V)22 b Fx(\))27 b Fu(3)i Ft(a)e Fu(7!)1488 3524 y Ff(P)1593 3628 y Fq(\033)r Fr(2)p Fq(S)1725 3636 y Fk(n)1789 3599 y Ft(\033)t Fx(\()p Ft(a)p Fx(\))h Fu(2)g Ft(T)2168 3563 y Fq(n)2214 3599 y Fx(\()p Ft(V)22 b Fx(\))32 b(is)g(a)h(linear)e(map.)149 3739 y(b\))i(Sho)m(w)h(that)e Fu(S)40 b Fx(has)33 b(its)f(image)f(in) 1565 3714 y(^)1549 3739 y Ft(S)1615 3703 y Fq(n)1662 3739 y Fx(\()p Ft(V)21 b Fx(\).)149 3867 y(c\))33 b(Sho)m(w)h(that)e (Im\()p Fu(S)7 b Fx(\))28 b(=)1140 3841 y(^)1123 3867 y Ft(S)1189 3830 y Fq(n)1236 3867 y Fx(\()p Ft(V)22 b Fx(\))32 b(if)g Ft(n)p Fx(!)g(is)g(in)m(v)m(ertible)g(in)g Fp(K)18 b Fx(.)149 3995 y(d\))37 b(Sho)m(w)h(that)772 3970 y(^)756 3995 y Ft(S)822 3959 y Fq(n)869 3995 y Fx(\()p Ft(V)21 b Fx(\))35 b Ft(,)-17 b Fu(!)35 b Ft(T)1274 3959 y Fq(n)1321 3995 y Fx(\()p Ft(V)21 b Fx(\))1540 3938 y Fq(\027)1510 3995 y Fu(!)35 b Ft(S)1711 3959 y Fq(n)1758 3995 y Fx(\()p Ft(V)21 b Fx(\))37 b(is)f(an)h(isomorphism)d(if)i Ft(n)p Fx(!)h(is)g(in)m(v)m(ertible)f(in)0 4111 y Fp(K)63 b Fx(and)39 b Ft(\027)45 b Fx(:)39 b Ft(T)543 4075 y Fq(n)590 4111 y Fx(\()p Ft(V)21 b Fx(\))39 b Fu(\000)-60 b(!)39 b Ft(S)1005 4075 y Fq(n)1052 4111 y Fx(\()p Ft(V)21 b Fx(\))40 b(is)e(the)i(restriction)e(of)h Ft(\027)45 b Fx(:)39 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))39 b Fu(\000)-60 b(!)39 b Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\),)41 b(the)f(symmetric)0 4227 y(algebra.)149 4343 y(4.)55 b(A)37 b(tensor)g Ft(a)e Fu(2)g Ft(T)945 4307 y Fq(n)991 4343 y Fx(\()p Ft(V)22 b Fx(\))36 b(is)g(called)f Fs(an)k(antisymmetric)f(tensor)e Fx(if)f Ft(\033)t Fx(\()p Ft(a)p Fx(\))g(=)f Ft(")p Fx(\()p Ft(\033)t Fx(\))p Ft(a)i Fx(for)g(all)0 4471 y Ft(\033)i Fu(2)d Ft(S)254 4486 y Fq(n)337 4471 y Fx(where)j Ft(")p Fx(\()p Ft(\033)t Fx(\))d(is)h(the)h(sign)f(of)g(the)h(p)s(erm)m (utation)e Ft(\033)t Fx(.)54 b(Let)2509 4446 y(^)2486 4471 y Ft(E)2564 4435 y Fq(n)2611 4471 y Fx(\()p Ft(V)21 b Fx(\))37 b(b)s(e)f(the)h(subspace)h(of)0 4587 y(an)m(tisymmetric)31 b(tensors)j(in)e Ft(T)1157 4551 y Fq(n)1203 4587 y Fx(\()p Ft(V)22 b Fx(\).)149 4703 y(a\))33 b(Sho)m(w)g(that)f Fu(E)37 b Fx(:)27 b Ft(T)952 4667 y Fq(n)999 4703 y Fx(\()p Ft(V)22 b Fx(\))27 b Fu(3)h Ft(a)g Fu(7!)1481 4629 y Ff(P)1587 4733 y Fq(\033)r Fr(2)p Fq(S)1719 4741 y Fk(n)1783 4703 y Ft(")p Fx(\()p Ft(\033)t Fx(\))p Ft(\033)t Fx(\()p Ft(a)p Fx(\))f Fu(2)h Ft(T)2342 4667 y Fq(n)2389 4703 y Fx(\()p Ft(V)21 b Fx(\))33 b(is)f(a)g(linear)f(map.)149 4844 y(b\))i(Sho)m(w)h(that)e Fu(E)41 b Fx(has)33 b(its)f(image)f(in) 1565 4818 y(^)1542 4844 y Ft(E)1620 4807 y Fq(n)1668 4844 y Fx(\()p Ft(V)21 b Fx(\).)149 4971 y(c\))33 b(Sho)m(w)h(that)e (Im\()p Fu(E)9 b Fx(\))27 b(=)1140 4946 y(^)1117 4971 y Ft(E)1195 4935 y Fq(n)1242 4971 y Fx(\()p Ft(V)22 b Fx(\))32 b(if)f Ft(n)p Fx(!)i(is)f(in)m(v)m(ertible)g(in)g Fp(K)18 b Fx(.)149 5099 y(d\))28 b(Sho)m(w)h(that)751 5074 y(^)728 5099 y Ft(E)806 5063 y Fq(n)853 5099 y Fx(\()p Ft(V)22 b Fx(\))27 b Ft(,)-17 b Fu(!)28 b Ft(T)1244 5063 y Fq(n)1291 5099 y Fx(\()p Ft(V)21 b Fx(\))1503 5043 y Fq(\027)1473 5099 y Fu(!)27 b Ft(E)1678 5063 y Fq(n)1725 5099 y Fx(\()p Ft(V)22 b Fx(\))27 b(is)h(an)f(isomorphism)f(if)g Ft(n)p Fx(!)i(is)g(in)m(v)m(ertible)f(in)g Fp(K)0 5216 y Fx(and)k Ft(\027)j Fx(:)28 b Ft(T)396 5179 y Fq(n)442 5216 y Fx(\()p Ft(V)22 b Fx(\))27 b Fu(\000)-60 b(!)28 b Ft(E)847 5179 y Fq(n)894 5216 y Fx(\()p Ft(V)22 b Fx(\))30 b(is)g(the)h(restriction)f(of)g Ft(\027)k Fx(:)28 b Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))27 b Fu(\000)-60 b(!)28 b Ft(E)6 b Fx(\()p Ft(V)22 b Fx(\),)31 b(the)g(exterior)f(algebra.)p eop %%Page: 26 26 26 25 bop 0 236 a Fo(26)1456 b(8.)77 b(TOOLBO)n(X)149 425 y Fv(De\014nition)37 b(8.5.19.)49 b Fx(Let)43 b Ft(A)f Fx(b)s(e)h(a)f Fp(K)18 b Fx(-algebra.)78 b(A)43 b Fs(left)h Ft(A)p Fs(-mo)-5 b(dule)42 b Fx(is)g(a)g Fp(K)18 b Fx(-mo)s(dule)46 b Ft(M)0 541 y Fx(together)33 b(with)f(a)g(homomorphism)e Ft(\026)1436 556 y Fq(M)1542 541 y Fx(:)e Ft(A)22 b Fu(\012)h Ft(M)39 b Fu(\000)-60 b(!)27 b Ft(M)10 b Fx(,)34 b(suc)m(h)g(that)e (the)h(diagrams)1309 1205 y Ft(A)22 b Fu(\012)h Ft(M)588 b(M)p 1637 1177 521 4 v 2075 1175 a Fl(-)1876 1223 y Fq(\026)1211 718 y Ft(A)23 b Fu(\012)f Ft(A)h Fu(\012)f Ft(M)394 b(A)22 b Fu(\012)h Ft(M)p 1735 689 326 4 v 1978 687 a Fl(-)1814 656 y Fn(id)11 b Fr(\012)p Fq(\026)p 1457 1107 4 351 v 1459 1107 a Fl(?)1247 951 y Fr(r\012)p Fn(id)p 2237 1107 V 2239 1107 a Fl(?)2278 939 y Fq(\026)0 1354 y Fx(and)1224 1470 y Ft(M)1356 1443 y Fu(\030)1357 1474 y Fx(=)1462 1470 y Fp(K)45 b Fu(\012)23 b Ft(M)321 b(A)22 b Fu(\012)h Ft(M)p 1794 1442 254 4 v 1965 1440 a Fl(-)1845 1408 y Fq(\021)r Fr(\012)p Fn(id)2174 1962 y Ft(M)p 2224 1859 4 351 v 2226 1859 a Fl(?)2265 1692 y Fq(\026)1801 1707 y Fn(id)1388 1550 y Fl(H)1471 1591 y(H)1554 1633 y(H)1637 1674 y(H)1720 1716 y(H)1803 1757 y(H)1886 1799 y(H)1969 1840 y(H)2007 1859 y(H)-83 b(j)0 2064 y Fx(comm)m(ute.)149 2180 y(Let)323 2195 y Fq(A)380 2180 y Ft(M)43 b Fx(and)706 2195 y Fq(A)763 2180 y Ft(N)f Fx(b)s(e)32 b Ft(A)p Fx(-mo)s(dules)e(and)i(let)f Ft(f)38 b Fx(:)28 b Ft(M)38 b Fu(\000)-60 b(!)28 b Ft(N)42 b Fx(b)s(e)32 b(a)g Fp(K)17 b Fx(-linear)36 b(map.)42 b(The)33 b(map)0 2297 y Ft(f)43 b Fx(is)32 b(called)g(a)g Fs(homomorphism)h(of)h (mo)-5 b(dules)32 b Fx(if)f(the)i(diagram)1313 2943 y Ft(A)22 b Fu(\012)h Ft(N)605 b(N)p 1625 2915 537 4 v 2079 2913 a Fl(-)1844 2961 y Fq(\026)1886 2972 y Fk(N)1305 2455 y Ft(A)22 b Fu(\012)h Ft(M)588 b(M)p 1633 2427 521 4 v 2071 2425 a Fl(-)1839 2393 y Fq(\026)1881 2404 y Fk(M)p 1453 2845 4 351 v 1455 2845 a Fl(?)1284 2686 y Fn(1)p Fr(\012)p Fq(f)p 2233 2845 V 2235 2845 a Fl(?)2274 2686 y Fq(f)0 3084 y Fx(comm)m(utes.)149 3200 y(The)23 b(left)e Ft(A)p Fx(-mo)s(dules)f(and)i(their)f(homomorphisms)f(form)g (the)j Fs(c)-5 b(ate)g(gory)2840 3215 y Fq(A)2897 3200 y Fu(M)25 b Fs(of)g Ft(A)p Fs(-mo)-5 b(dules)p Fx(.)149 3374 y Fv(Problem)36 b(8.5.10.)49 b Fx(Sho)m(w)37 b(that)e(an)g(ab)s (elian)f(group)h Ft(M)47 b Fx(is)35 b(a)g(left)f(mo)s(dule)h(o)m(v)m (er)h(the)g(ring)0 3490 y Ft(A)c Fx(if)f(and)h(only)g(if)f Ft(M)43 b Fx(is)32 b(a)f Fp(K)18 b Fx(-mo)s(dule)36 b(and)c(an)g Ft(A)p Fx(-mo)s(dule)f(in)g(the)i(sense)h(of)d(De\014nition)g(8.5.19.)p eop %%Page: 27 27 27 26 bop 1474 236 a Fo(6.)76 b(CO)n(ALGEBRAS)1371 b(27)1441 425 y Fv(6.)94 b(Coalgebras)149 600 y(De\014nition)37 b(8.6.1.)49 b Fx(A)34 b Fp(K)18 b Fs(-c)-5 b(o)g(algebr)g(a)39 b Fx(is)34 b(a)g Fp(K)18 b Fx(-mo)s(dule)38 b Ft(C)k Fx(together)35 b(with)f(a)g Fs(c)-5 b(omultiplic)g(a-)0 716 y(tion)32 b Fx(or)h Fs(diagonal)e Fx(\001)d(:)g Ft(C)34 b Fu(\000)-60 b(!)28 b Ft(C)h Fu(\012)23 b Ft(C)39 b Fx(that)32 b(is)h(coasso)s(ciativ)m(e:)1222 1373 y Ft(C)d Fu(\012)22 b Ft(C)412 b(C)29 b Fu(\012)23 b Ft(C)29 b Fu(\012)22 b Ft(C)p 1527 1345 348 4 v 1792 1343 a Fl(-)1608 1409 y Fn(id)12 b Fr(\012)p Fn(\001)1322 886 y Ft(C)610 b(C)30 b Fu(\012)22 b Ft(C)p 1428 858 546 4 v 1891 856 a Fl(-)1671 837 y Fn(\001)p 1359 1275 4 351 v 1360 1275 a Fl(?)1262 1122 y Fn(\001)p 2139 1275 V 2140 1275 a Fl(?)2179 1119 y Fn(\001)p Fr(\012)p Fn(id)0 1541 y Fx(and)33 b(a)f Fs(c)-5 b(ounit)33 b Fx(or)f Fs(augmentation)g Ft(\017)c Fx(:)f Ft(C)35 b Fu(\000)-60 b(!)28 b Fp(K)17 b Fx(:)1113 1711 y Ft(C)805 b(C)29 b Fu(\012)23 b Ft(C)p 1218 1683 741 4 v 1876 1681 a Fl(-)1560 1662 y Fn(\001)p 1150 2100 4 351 v 1151 2100 a Fl(?)1053 1947 y Fn(\001)p 2124 2100 V 2126 2100 a Fl(?)2165 1944 y Fn(id)12 b Fr(\012)p Fq(\017)1013 2199 y Ft(C)29 b Fu(\012)23 b Ft(C)383 b Fp(K)46 b Fu(\012)22 b Ft(C)1969 2171 y Fu(\030)1970 2203 y Fx(=)2074 2199 y Ft(C)2179 2171 y Fu(\030)2180 2203 y Fx(=)2284 2199 y Ft(C)29 b Fu(\012)23 b Fp(K)17 b Ft(:)p 1318 2171 319 4 v 1554 2169 a Fl(-)1406 2234 y Fq(\017)p Fr(\012)p Fn(id)1701 1948 y(id)1288 1791 y Fl(H)1371 1832 y(H)1454 1874 y(H)1537 1915 y(H)1620 1957 y(H)1703 1998 y(H)1786 2040 y(H)1869 2081 y(H)1907 2100 y(H)-83 b(j)0 2366 y Fx(A)33 b Fp(K)17 b Fx(-coalgebra)37 b Ft(C)j Fx(is)32 b Fs(c)-5 b(o)g(c)g(ommutative)31 b Fx(if)h(the)h(follo)m(wing)d(diagram)g(comm)m(utes)1418 2992 y Ft(C)g Fu(\012)22 b Ft(C)219 b(C)29 b Fu(\012)23 b Ft(C)p 1723 2965 155 4 v 1795 2963 a Fl(-)1781 2944 y Fq(\034)1762 2509 y Ft(C)1560 2742 y Fn(\001)1724 2626 y Fl(\001)1683 2709 y(\001)1641 2792 y(\001)1600 2875 y(\001)1590 2894 y(\001)-42 b(\013)1981 2742 y Fn(\001)1834 2626 y Fl(A)1876 2709 y(A)1917 2792 y(A)1959 2875 y(A)1968 2894 y(A)g(U)149 3127 y Fx(Let)38 b Ft(C)44 b Fx(and)38 b Ft(D)i Fx(b)s(e)d Fp(K)18 b Fx(-coalgebras.)62 b(A)38 b Fs(homomorphism)f(of)i(c)-5 b(o)g(algebr)g(as)36 b Ft(f)46 b Fx(:)36 b Ft(C)43 b Fu(\000)-60 b(!)35 b Ft(D)40 b Fx(is)d(a)0 3243 y Fp(K)17 b Fx(-linear)37 b(map)32 b(suc)m(h)i(that)e(the)h(follo)m(wing)d(diagrams)h(comm)m(ute:)1368 3914 y Ft(C)e Fu(\012)22 b Ft(C)310 b(D)25 b Fu(\012)d Ft(D)p 1672 3886 245 4 v 1834 3884 a Fl(-)1725 3950 y Fq(f)7 b Fr(\012)p Fq(f)1467 3431 y Ft(C)512 b(D)p 1573 3398 448 4 v 1938 3396 a Fl(-)1775 3365 y Fq(f)p 1504 3816 4 351 v 1506 3816 a Fl(?)1356 3658 y Fn(\001)1415 3669 y Fk(C)p 2089 3816 V 2091 3816 a Fl(?)2130 3658 y Fn(\001)2189 3669 y Fk(D)0 4088 y Fx(and)1753 4703 y Fp(K)1530 4434 y Fq(\017)1559 4445 y Fk(C)1582 4333 y Fl(A)1624 4416 y(A)1665 4499 y(A)1707 4582 y(A)1716 4601 y(A)-42 b(U)1973 4434 y Fq(\017)2002 4445 y Fk(D)1960 4333 y Fl(\001)1918 4416 y(\001)1877 4499 y(\001)1835 4582 y(\001)1826 4601 y(\001)g(\013)1510 4216 y Ft(C)400 b(D)s(:)p 1615 4183 337 4 v 1869 4181 a Fl(-)1762 4149 y Fq(f)149 4848 y Fv(Remark)37 b(8.6.2.)50 b Fx(Ob)m(viously)28 b(the)h(comp)s(osition)d(of)i(t)m(w)m(o)h(homomorphisms)d(of)h (coalgebras)0 4964 y(is)39 b(again)f(a)h(homomorphism)e(of)i (coalgebras.)63 b(F)-8 b(urthermore)39 b(the)h(iden)m(tit)m(y)f(map)f (is)h(a)g(homo-)0 5080 y(morphism)j(of)h(coalgebras.)75 b(Hence)45 b(the)f Fp(K)17 b Fx(-coalgebras)48 b(form)43 b(a)g(category)g Fp(K)18 b Fx(-)p Fv(Coalg)5 b Fx(.)76 b(The)0 5196 y(category)33 b(of)f(co)s(comm)m(utativ)m(e)f Fp(K)18 b Fx(-coalgebras)37 b(will)31 b(b)s(e)i(denoted)g(b)m(y)h Fp(K)17 b Fx(-)p Fv(cCoalg)5 b Fx(.)p eop %%Page: 28 28 28 27 bop 0 236 a Fo(28)1456 b(8.)77 b(TOOLBO)n(X)149 425 y Fv(Problem)36 b(8.6.11.)49 b Fx(1.)42 b(Sho)m(w)27 b(that)f Ft(V)32 b Fu(\012)10 b Ft(V)1797 389 y Fr(\003)1863 425 y Fx(is)26 b(a)g(coalgebra)g(for)g(ev)m(ery)i(\014nite)e (dimensional)0 541 y(v)m(ector)48 b(space)g Ft(V)68 b Fx(o)m(v)m(er)48 b(a)f(\014eld)f Fp(K)70 b Fx(if)46 b(the)h(com)m (ultiplication)c(is)j(de\014ned)i(b)m(y)g(\001\()p Ft(v)36 b Fu(\012)c Ft(v)3368 505 y Fr(\003)3407 541 y Fx(\))52 b(:=)0 583 y Ff(P)105 609 y Fq(n)105 687 y(i)p Fn(=1)240 658 y Ft(v)26 b Fu(\012)d Ft(v)464 622 y Fr(\003)460 682 y Fq(i)525 658 y Fu(\012)g Ft(v)672 673 y Fq(i)722 658 y Fu(\012)g Ft(v)873 622 y Fr(\003)944 658 y Fx(where)34 b Fu(f)p Ft(v)1323 673 y Fq(i)1351 658 y Fu(g)f Fx(and)f Fu(f)p Ft(v)1724 622 y Fr(\003)1720 682 y Fq(i)1763 658 y Fu(g)h Fx(are)f(dual)g(bases)i(of)e Ft(V)54 b Fx(resp.)34 b Ft(V)3015 622 y Fr(\003)3055 658 y Fx(.)149 774 y(2.)48 b(Sho)m(w)35 b(that)f(the)g(free)h Fp(K)17 b Fx(-mo)s(dules)39 b Fp(K)17 b Ft(X)48 b Fx(with)34 b(the)g(basis)g Ft(X)42 b Fx(and)34 b(the)g(com)m(ultiplication)0 890 y(\001\()p Ft(x)p Fx(\))28 b(=)g Ft(x)22 b Fu(\012)h Ft(x)33 b Fx(is)f(a)g (coalgebra.)43 b(What)32 b(is)h(the)g(counit?)43 b(Is)33 b(the)g(counit)f(unique?)149 1006 y(3.)43 b(Sho)m(w)32 b(that)e Fp(K)42 b Fu(\010)19 b Ft(V)53 b Fx(with)30 b(\001\(1\))e(=)f(1)19 b Fu(\012)g Fx(1,)31 b(\001\()p Ft(v)t Fx(\))c(=)h Ft(v)22 b Fu(\012)d Fx(1)f(+)h(1)f Fu(\012)h Ft(v)35 b Fx(de\014nes)d(a)f(coalgebra.)149 1123 y(4.)40 b(Let)24 b Ft(C)30 b Fx(and)23 b Ft(D)i Fx(b)s(e)f(coalgebras.)39 b(Then)25 b Ft(C)10 b Fu(\012)s Ft(D)25 b Fx(is)e(a)f(coalgebra)g(with)h(the)g(com)m(ultiplication)0 1239 y(\001)81 1254 y Fq(C)5 b Fr(\012)p Fq(D)283 1239 y Fx(:=)27 b(\(1)500 1254 y Fq(C)576 1239 y Fu(\012)17 b Ft(\034)28 b Fu(\012)17 b Fx(1)883 1254 y Fq(D)947 1239 y Fx(\)\(\001)1104 1254 y Fq(C)1180 1239 y Fu(\012)g Fx(\001)1355 1254 y Fq(D)1419 1239 y Fx(\))28 b(:)g Ft(C)23 b Fu(\012)17 b Ft(D)j Fu(\012)d Ft(C)23 b Fu(\012)17 b Ft(D)31 b Fu(\000)-60 b(!)27 b Ft(C)d 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Fn(\(1\))1993 1956 y Fu(\012)23 b Ft(c)2135 1971 y Fn(\(2\))2229 1956 y Ft(:)0 2138 y Fx(Observ)m(e)39 b(that)e(only)f(the)i(complete)e(expression)j(on)e (the)g(righ)m(t)g(hand)g(side)g(mak)m(es)h(sense,)i(not)0 2254 y(the)27 b(comp)s(onen)m(ts)g Ft(c)735 2270 y Fn(\(1\))856 2254 y Fx(or)g Ft(c)1012 2270 y Fn(\(2\))1132 2254 y Fx(whic)m(h)h(are)e Fs(not)h Fx(considered)g(as)g(families)d(of)i (elemen)m(ts)h(of)f Ft(C)7 b Fx(.)42 b(This)0 2371 y(notation)36 b(alone)g(do)s(es)h(not)g(help)g(m)m(uc)m(h)h(in)e(the)i(calculations)d (w)m(e)j(ha)m(v)m(e)g(to)f(p)s(erform)f(later)g(on.)0 2487 y(So)c(w)m(e)i(in)m(tro)s(duce)f(a)f(more)g(general)g(notation.) 149 2667 y Fv(De\014nition)37 b(8.6.3.)49 b Fx(\(Sw)m(eedler)35 b(Notation\))d(Let)j Ft(M)44 b Fx(b)s(e)34 b(an)g(arbitrary)f Fp(K)18 b Fx(-mo)s(dule)38 b(and)c Ft(C)0 2783 y Fx(b)s(e)f(a)f Fp(K)18 b Fx(-coalgebra.)48 b(Then)33 b(there)h(is)e(a)g(bijection)f(b) s(et)m(w)m(een)k(all)30 b(m)m(ultilinear)f(maps)1335 2949 y Ft(f)38 b Fx(:)28 b Ft(C)h Fu(\002)23 b Ft(:)17 b(:)g(:)k Fu(\002)i Ft(C)35 b Fu(\000)-60 b(!)27 b Ft(M)0 3116 y Fx(and)33 b(all)d(linear)h(maps)1312 3242 y Ft(f)1371 3201 y Fr(0)1422 3242 y Fx(:)d Ft(C)h Fu(\012)23 b Ft(:)17 b(:)g(:)k Fu(\012)i Ft(C)34 b Fu(\000)-60 b(!)28 b Ft(M)5 b(:)0 3388 y Fx(These)34 b(maps)f(are)f(asso)s(ciated)h(to)f(eac)m(h)h (other)g(b)m(y)h(the)f(form)m(ula)1108 3555 y Ft(f)11 b Fx(\()p Ft(c)1247 3570 y Fn(1)1286 3555 y Ft(;)17 b(:)g(:)g(:)32 b(;)17 b(c)1563 3570 y Fq(n)1610 3555 y Fx(\))27 b(=)h Ft(f)1838 3514 y Fr(0)1861 3555 y Fx(\()p Ft(c)1941 3570 y Fn(1)2002 3555 y Fu(\012)23 b Ft(:)17 b(:)g(:)22 b Fu(\012)g Ft(c)2380 3570 y Fq(n)2427 3555 y Fx(\))p Ft(:)0 3721 y Fx(F)-8 b(or)32 b Ft(c)27 b Fu(2)i Ft(C)39 b Fx(w)m(e)34 b(de\014ne)1055 3753 y Ff(X)1216 3848 y Ft(f)11 b Fx(\()p Ft(c)1355 3863 y Fn(\(1\))1448 3848 y Ft(;)17 b(:)g(:)g(:)33 b(;)17 b(c)1726 3863 y Fn(\()p Fq(n)p Fn(\))1827 3848 y Fx(\))28 b(:=)g Ft(f)2083 3807 y Fr(0)2106 3848 y Fx(\(\001)2225 3807 y Fq(n)p Fr(\000)p Fn(1)2362 3848 y Fx(\()p Ft(c)p Fx(\)\))p Ft(;)0 4020 y Fx(where)38 b(\001)367 3984 y Fq(n)p Fr(\000)p Fn(1)541 4020 y Fx(denotes)f(the)g Ft(n)25 b Fu(\000)g Fx(1-fold)35 b(application)f(of)i(\001,)i(for)e(example)g (\001)2921 3984 y Fq(n)p Fr(\000)p Fn(1)3092 4020 y Fx(=)f(\(\001)25 b Fu(\012)g Fx(1)g Fu(\012)0 4136 y Ft(:)17 b(:)g(:)22 b Fu(\012)g Fx(1\))g Fu(\016)g Fx(\(\001)g Fu(\012)h Fx(1\))f Fu(\016)g Fx(\001.)149 4252 y(In)33 b(particular)e(w)m(e)j (obtain)d(for)h(the)h(bilinear)e(map)g Fu(\012)d Fx(:)g Ft(C)h Fu(\002)23 b Ft(C)35 b Fu(3)28 b Fx(\()p Ft(c;)17 b(d)p Fx(\))27 b Fu(7!)g Ft(c)22 b Fu(\012)h Ft(d)k Fu(2)h Ft(C)h Fu(\012)23 b Ft(C)1344 4346 y Ff(X)1504 4440 y Ft(c)1546 4456 y Fn(\(1\))1663 4440 y Fu(\012)f Ft(c)1804 4456 y Fn(\(2\))1926 4440 y Fx(=)28 b(\001\()p Ft(c)p Fx(\))p Ft(;)0 4633 y Fx(and)33 b(for)f(the)h(m)m(ultilinear)c(map)i Fu(\012)1296 4597 y Fn(2)1364 4633 y Fx(:)d Ft(C)h Fu(\002)22 b Ft(C)30 b Fu(\002)22 b Ft(C)35 b Fu(\000)-60 b(!)28 b Ft(C)h Fu(\012)22 b Ft(C)29 b Fu(\012)23 b Ft(C)722 4721 y Ff(X)882 4815 y Ft(c)924 4831 y Fn(\(1\))1041 4815 y Fu(\012)f Ft(c)1182 4831 y Fn(\(2\))1299 4815 y Fu(\012)h Ft(c)1441 4831 y Fn(\(3\))1563 4815 y Fx(=)k(\(\001)22 b Fu(\012)h Fx(1\)\001\()p Ft(c)p Fx(\))28 b(=)f(\(1)22 b Fu(\012)h Fx(\001\)\001\()p Ft(c)p Fx(\))p Ft(:)149 5011 y Fx(With)32 b(this)h(notation)e(one)i(v)m(eri\014es)g(easily)570 5099 y Ff(X)731 5194 y Ft(c)773 5209 y Fn(\(1\))889 5194 y Fu(\012)23 b Ft(:)17 b(:)g(:)22 b Fu(\012)g Fx(\001\()p Ft(c)1386 5209 y Fn(\()p Fq(i)p Fn(\))1470 5194 y Fx(\))g Fu(\012)g Ft(:)17 b(:)g(:)22 b Fu(\012)h Ft(c)1908 5209 y Fn(\()p Fq(n)p Fn(\))2037 5194 y Fx(=)2141 5099 y Ff(X)2301 5194 y Ft(c)2343 5209 y Fn(\(1\))2460 5194 y Fu(\012)f Ft(:)17 b(:)g(:)22 b Fu(\012)g Ft(c)2837 5209 y Fn(\()p Fq(n)p Fn(+1\))p eop %%Page: 29 29 29 28 bop 1474 236 a Fo(6.)76 b(CO)n(ALGEBRAS)1371 b(29)0 425 y Fx(and)399 440 y Ff(P)521 515 y Ft(c)563 530 y Fn(\(1\))679 515 y Fu(\012)23 b Ft(:)17 b(:)g(:)22 b Fu(\012)g Ft(\017)p Fx(\()p Ft(c)1134 530 y Fn(\()p Fq(i)p Fn(\))1218 515 y Fx(\))g Fu(\012)g Ft(:)17 b(:)g(:)22 b Fu(\012)h Ft(c)1656 530 y Fn(\()p Fq(n)p Fn(\))1840 515 y Fx(=)1944 440 y Ff(P)2066 515 y Ft(c)2108 530 y Fn(\(1\))2224 515 y Fu(\012)g Ft(:)17 b(:)g(:)k Fu(\012)i Fx(1)f Fu(\012)h Ft(:)17 b(:)g(:)k Fu(\012)i Ft(c)3009 530 y Fn(\()p Fq(n)p Fr(\000)p Fn(1\))1840 631 y Fx(=)1944 556 y Ff(P)2066 631 y Ft(c)2108 647 y Fn(\(1\))2224 631 y Fu(\012)g Ft(:)17 b(:)g(:)k Fu(\012)i Ft(c)2602 647 y Fn(\()p Fq(n)p Fr(\000)p Fn(1\))149 764 y Fx(This)39 b(notation)e(and)i(its)f(application)e(to)i(m)m(ultilinear)d(maps)j (will)e(also)i(b)s(e)g(used)i(in)e(more)0 880 y(general)32 b(con)m(texts)i(lik)m(e)e(como)s(dules.)149 1054 y Fv(Prop)s(osition)k (8.6.4.)49 b Fs(L)-5 b(et)27 b Ft(C)33 b Fs(b)-5 b(e)26 b(a)h(c)-5 b(o)g(algebr)g(a)25 b(and)h Ft(A)g Fs(an)g(algebr)-5 b(a.)41 b(Then)26 b(the)g(c)-5 b(omp)g(osition)0 1170 y Ft(f)33 b Fu(\003)22 b Ft(g)31 b Fx(:=)c Fu(r)444 1185 y Fq(A)501 1170 y Fx(\()p Ft(f)33 b Fu(\012)23 b Ft(g)t Fx(\)\001)890 1185 y Fr(C)969 1170 y Fs(de\014nes)34 b(a)h(multiplic)-5 b(ation)608 1327 y Fx(Hom)o(\()p Ft(C)r(;)17 b(A)p Fx(\))22 b Fu(\012)g Fx(Hom\()p Ft(C)r(;)17 b(A)p Fx(\))27 b Fu(3)h Ft(f)k Fu(\012)23 b Ft(g)31 b Fu(7!)c Ft(f)33 b Fu(\003)22 b Ft(g)31 b Fu(2)d Fx(Hom)o(\()p Ft(C)r(;)17 b(A)p Fx(\))p Ft(;)0 1484 y Fs(such)43 b(that)g Fx(Hom)o(\()p Ft(C)r(;)17 b(A)p Fx(\))43 b Fs(b)-5 b(e)g(c)g(omes)41 b(an)i(algebr)-5 b(a.)68 b(The)43 b(unit)g(element)f(is)h(given)f(by)i Fp(K)66 b Fu(3)43 b Ft(\013)g Fu(7!)0 1600 y Fx(\()p Ft(c)28 b Fu(7!)f Ft(\021)t Fx(\()p Ft(\013)q(\017)p Fx(\()p Ft(c)p Fx(\)\)\))g Fu(2)h Fx(Hom)o(\()p Ft(C)r(;)17 b(A)p Fx(\))p Fs(.)149 1774 y Fi(Pr)n(oof.)42 b Fx(The)35 b(m)m(ultiplication)30 b(of)j(Hom\()p Ft(C)r(;)17 b(A)p Fx(\))33 b(ob)m(viously)h(is)f(a)h(bilinear)e(map.)47 b(The)35 b(m)m(ul-)0 1890 y(tiplication)29 b(is)k(asso)s(ciativ)m(e)f (since)h(\()p Ft(f)g Fu(\003)22 b Ft(g)t Fx(\))g Fu(\003)g Ft(h)28 b Fx(=)g Fu(r)1935 1905 y Fq(A)1992 1890 y Fx(\(\()p Fu(r)2151 1905 y Fq(A)2208 1890 y Fx(\()p Ft(f)33 b Fu(\012)23 b Ft(g)t Fx(\)\001)2597 1905 y Fq(C)2656 1890 y Fx(\))f Fu(\012)h Ft(h)p Fx(\)\001)2991 1905 y Fq(C)3078 1890 y Fx(=)28 b Fu(r)3265 1905 y Fq(A)3322 1890 y Fx(\()p Fu(r)3443 1905 y Fq(A)3523 1890 y Fu(\012)0 2007 y Fx(1\)\(\()p Ft(f)22 b Fu(\012)12 b Ft(g)t Fx(\))g Fu(\012)g Ft(h)p Fx(\)\(\001)725 2022 y Fq(C)796 2007 y Fu(\012)g Fx(1\)\001)1053 2022 y Fq(C)1140 2007 y Fx(=)28 b Fu(r)1327 2022 y Fq(A)1384 2007 y Fx(\(1)12 b Fu(\012)g(r)1655 2022 y Fq(A)1712 2007 y Fx(\)\()p Ft(f)22 b Fu(\012)12 b Fx(\()p Ft(g)j Fu(\012)d Ft(h)p Fx(\)\)\(1)g Fu(\012)g Fx(\001)2537 2022 y Fq(C)2597 2007 y Fx(\)\001)2716 2022 y Fq(C)2803 2007 y Fx(=)28 b Fu(r)2990 2022 y Fq(A)3047 2007 y Fx(\()p Ft(f)22 b Fu(\012)12 b Fx(\()p Fu(r)3365 2022 y Fq(A)3422 2007 y Fx(\()p Ft(g)j Fu(\012)0 2123 y Ft(h)p Fx(\)\001)175 2138 y Fq(C)235 2123 y Fx(\)\)\001)392 2138 y Fq(C)481 2123 y Fx(=)31 b Ft(f)j Fu(\003)23 b Fx(\()p Ft(g)j Fu(\003)e Ft(h)p Fx(\).)48 b(F)-8 b(urthermore)34 b(it)f(is)h(unitary)g(with)g (unit)f(1)2689 2138 y Fn(Hom)o(\()p Fq(C)q(;A)p Fn(\))3049 2123 y Fx(=)e Ft(\021)3204 2138 y Fq(A)3261 2123 y Ft(\017)3300 2138 y Fq(C)3394 2123 y Fx(since)0 2239 y Ft(\021)48 2254 y Fq(A)105 2239 y Ft(\017)144 2254 y Fq(C)225 2239 y Fu(\003)22 b Ft(f)38 b Fx(=)28 b Fu(r)570 2254 y Fq(A)627 2239 y Fx(\()p Ft(\021)713 2254 y Fq(A)770 2239 y Ft(\017)809 2254 y Fq(C)890 2239 y Fu(\012)22 b Ft(f)11 b Fx(\)\001)1167 2254 y Fq(C)1254 2239 y Fx(=)28 b Fu(r)1441 2254 y Fq(A)1498 2239 y Fx(\()p Ft(\021)1584 2254 y Fq(A)1663 2239 y Fu(\012)22 b Fx(1)1811 2254 y Fq(A)1868 2239 y Fx(\)\(1)1993 2254 y Fq(K)2082 2239 y Fu(\012)h Ft(f)11 b Fx(\)\()p Ft(\017)2356 2254 y Fq(C)2436 2239 y Fu(\012)23 b Fx(1)2585 2254 y Fq(C)2644 2239 y Fx(\)\001)2763 2254 y Fq(C)2850 2239 y Fx(=)k Ft(f)43 b Fx(and)33 b(similarly)0 2355 y Ft(f)g Fu(\003)22 b Ft(\021)201 2370 y Fq(A)258 2355 y Ft(\017)297 2370 y Fq(C)384 2355 y Fx(=)28 b Ft(f)11 b Fx(.)p 3529 2355 4 66 v 3533 2293 59 4 v 3533 2355 V 3591 2355 4 66 v 149 2529 a Fv(De\014nition)37 b(8.6.5.)49 b Fx(The)27 b(m)m(ultiplication)22 b Fu(\003)27 b Fx(:)h(Hom)o(\()p Ft(C)r(;)17 b(A)p Fx(\))9 b Fu(\012)g Fx(Hom\()p Ft(C)r(;)17 b(A)p Fx(\))27 b Fu(\000)-60 b(!)28 b Fx(Hom)o(\()p Ft(C)r(;)17 b(A)p Fx(\))0 2646 y(is)32 b(called)f Fs(c)-5 b(onvolution)p Fx(.)149 2820 y Fv(Corollary)36 b(8.6.6.)50 b Fs(L)-5 b(et)46 b Ft(C)52 b Fs(b)-5 b(e)45 b(a)h Fp(K)17 b Fs(-c)-5 b(o)g(algebr)g(a.)82 b(Then)45 b Ft(C)2426 2783 y Fr(\003)2513 2820 y Fx(=)i(Hom)2839 2835 y Fq(K)2908 2820 y Fx(\()p Ft(C)r(;)17 b Fp(K)f Fx(\))52 b Fs(is)45 b(an)h Fp(K)17 b Fs(-)0 2936 y(algebr)-5 b(a.)149 3110 y Fi(Pr)n(oof.)42 b Fx(Use)34 b(that)e Fp(K)56 b Fx(itself)32 b(is)g(a)g Fp(K)17 b Fx(-algebra.)p 3529 3110 V 3533 3047 59 4 v 3533 3110 V 3591 3110 4 66 v 149 3284 a Fv(Remark)37 b(8.6.7.)50 b Fx(If)33 b(w)m(e)i(write)f(the)g(ev)-5 b(aluation)32 b(as)i Ft(C)2216 3248 y Fr(\003)2279 3284 y Fu(\012)23 b Ft(C)37 b Fu(3)30 b Ft(a)24 b Fu(\012)f Ft(c)30 b Fu(7!)f(h)p Ft(a;)17 b(c)p Fu(i)30 b(2)g Fp(K)57 b Fx(then)0 3400 y(an)34 b(elemen)m(t)g Ft(a)c Fu(2)h Ft(C)754 3364 y Fr(\003)827 3400 y Fx(is)j(completely)f(determined)h(b) m(y)h(the)f(v)-5 b(alues)34 b(of)g Fu(h)p Ft(a;)17 b(c)p Fu(i)33 b Fx(for)g(all)f Ft(c)f Fu(2)f Ft(C)7 b Fx(.)48 b(So)0 3516 y(the)33 b(pro)s(duct)g(of)f Ft(a)h Fx(and)f Ft(b)h Fx(in)f Ft(C)1183 3480 y Fr(\003)1255 3516 y Fx(is)g(uniquely)h (determined)f(b)m(y)i(the)f(form)m(ula)874 3689 y Fu(h)p Ft(a)22 b Fu(\003)g Ft(b;)17 b(c)p Fu(i)28 b Fx(=)f Fu(h)p Ft(a)22 b Fu(\012)h Ft(b;)17 b Fx(\001\()p Ft(c)p Fx(\))p Fu(i)28 b Fx(=)2021 3595 y Ff(X)2182 3689 y Ft(a)p Fx(\()p Ft(c)2313 3705 y Fn(\(1\))2407 3689 y Fx(\))p Ft(b)p Fx(\()p Ft(c)2566 3705 y Fn(\(2\))2661 3689 y Fx(\))p Ft(:)0 3862 y Fx(The)34 b(unit)d(elemen)m(t)i(of)f Ft(C)955 3826 y Fr(\003)1027 3862 y Fx(is)g Ft(\017)c Fu(2)g Ft(C)1363 3826 y Fr(\003)1402 3862 y Fx(.)149 4036 y Fv(Lemma)38 b(8.6.8.)49 b Fs(L)-5 b(et)39 b Fp(K)62 b Fs(b)-5 b(e)38 b(a)g(\014eld)g(and)g Ft(A)h Fs(b)-5 b(e)38 b(a)h(\014nite)f (dimensional)f Fp(K)17 b Fs(-algebr)-5 b(a.)61 b(Then)0 4152 y Ft(A)73 4116 y Fr(\003)140 4152 y Fx(=)28 b(Hom)447 4167 y Fq(K)515 4152 y Fx(\()p Ft(A;)17 b Fp(K)h Fx(\))40 b Fs(is)35 b(a)g Fp(K)17 b Fs(-c)-5 b(o)g(algebr)g(a.)149 4326 y Fi(Pr)n(oof.)42 b Fx(De\014ne)34 b(the)f(com)m(ultiplication)28 b(on)k Ft(C)1935 4290 y Fr(\003)2007 4326 y Fx(b)m(y)1053 4517 y(\001)c(:)g Ft(A)1290 4475 y Fr(\003)1390 4460 y(r)1449 4437 y Fh(\003)1357 4517 y Fu(\000)-16 b(!)28 b Fx(\()p Ft(A)22 b Fu(\012)g Ft(A)p Fx(\))1889 4475 y Fr(\003)1957 4460 y Fn(can)2062 4437 y Fh(\000)p Fc(1)1970 4517 y Fu(\000)-16 b(!)42 b Ft(A)2246 4475 y Fr(\003)2307 4517 y Fu(\012)23 b Ft(A)2480 4475 y Fr(\003)2520 4517 y Ft(:)0 4674 y Fx(The)g(canonical)e(map)g(can)28 b(:)g Ft(A)1116 4637 y Fr(\003)1156 4674 y Fu(\012)q Ft(A)1307 4637 y Fr(\003)1374 4674 y Fu(\000)-60 b(!)28 b Fx(\()p Ft(A)q Fu(\012)q Ft(A)p Fx(\))1820 4637 y Fr(\003)1881 4674 y Fx(is)22 b(in)m(v)m(ertible,)h(since)g Ft(A)f Fx(is)g(\014nite)f(dimensional.)0 4790 y(By)35 b(a)f(diagrammatic)d (pro)s(of)j(or)g(b)m(y)h(calculation)d(with)i(elemen)m(ts)h(it)f(is)g (easy)h(to)f(sho)m(w)i(that)e Ft(A)3560 4754 y Fr(\003)0 4906 y Fx(b)s(ecomes)f(a)f Fp(K)18 b Fx(-coalgebra.)p 3529 4906 V 3533 4843 59 4 v 3533 4906 V 3591 4906 4 66 v 149 5080 a Fv(Remark)37 b(8.6.9.)50 b Fx(If)36 b Fp(K)59 b Fx(is)36 b(an)g(arbitrary)f(comm)m(utativ)m(e)g(ring,)i(then) f Ft(A)2875 5044 y Fr(\003)2949 5080 y Fx(=)d(Hom)3261 5095 y Fq(K)3330 5080 y Fx(\()p Ft(A;)17 b Fp(K)g Fx(\))0 5196 y(is)32 b(a)g Fp(K)18 b Fx(-coalgebra)37 b(if)31 b Ft(A)i Fx(is)f(a)g(\014nitely)g(generated)i(pro)5 b(jectiv)m(e)33 b Fp(K)18 b Fx(-mo)s(dule.)p eop %%Page: 30 30 30 29 bop 0 236 a Fo(30)1456 b(8.)77 b(TOOLBO)n(X)149 425 y Fv(Problem)36 b(8.6.12.)49 b Fx(Find)27 b(su\016cien)m(t)i (conditions)e(for)g(an)h(algebra)e Ft(A)i Fx(resp.)43 b(a)27 b(coalgebra)g Ft(C)0 541 y Fx(suc)m(h)34 b(that)e(Hom\()p Ft(A;)17 b(C)7 b Fx(\))32 b(b)s(ecomes)h(a)f(coalgebra)g(with)g(co-con) m(v)m(olution)f(as)i(com)m(ultiplication.)149 722 y Fv(De\014nition)k (8.6.10.)49 b Fx(Let)39 b Ft(C)46 b Fx(b)s(e)39 b(a)g Fp(K)18 b Fx(-coalgebra.)67 b(A)39 b Fs(left)i Ft(C)7 b Fs(-c)-5 b(omo)g(dule)38 b Fx(is)h(a)f Fp(K)18 b Fx(-mo)s(dule)0 838 y Ft(M)43 b Fx(together)33 b(with)f(a)h(homomorphism)c Ft(\016)1557 853 y Fq(M)1664 838 y Fx(:)f Ft(M)38 b Fu(\000)-60 b(!)28 b Ft(C)h Fu(\012)23 b Ft(M)10 b Fx(,)33 b(suc)m(h)h(that)f(the)g (diagrams)1209 1498 y Ft(C)c Fu(\012)22 b Ft(M)388 b(C)29 b Fu(\012)23 b Ft(C)29 b Fu(\012)23 b Ft(M)p 1541 1470 320 4 v 1778 1468 a Fl(-)1621 1534 y Fn(id)12 b Fr(\012)p Fq(\016)1308 1011 y Ft(M)587 b(C)29 b Fu(\012)22 b Ft(M)p 1441 983 519 4 v 1877 981 a Fl(-)1684 962 y Fq(\016)p 1359 1400 4 351 v 1360 1400 a Fl(?)1288 1248 y Fq(\016)p 2139 1400 V 2140 1400 a Fl(?)2179 1244 y Fn(\001)p Fr(\012)p Fn(id)0 1669 y Fx(and)1195 1755 y Ft(M)p 1246 2140 V 1248 2140 a Fl(?)1175 1988 y Fq(\016)1096 2238 y Ft(C)29 b Fu(\012)23 b Ft(M)552 b Fp(K)46 b Fu(\012)22 b Ft(M)2273 2211 y Fu(\030)2273 2243 y Fx(=)2378 2238 y Ft(M)5 b(:)p 1428 2210 485 4 v 1830 2208 a Fl(-)1599 2274 y Fq(\017)p Fr(\012)p Fn(id)1798 1988 y(id)1384 1830 y Fl(H)1467 1872 y(H)1550 1914 y(H)1633 1955 y(H)1716 1997 y(H)1799 2038 y(H)1882 2080 y(H)1965 2121 y(H)2003 2140 y(H)-83 b(j)0 2381 y Fx(comm)m(ute.)149 2497 y(Let)329 2461 y Fq(C)388 2497 y Ft(M)48 b Fx(and)725 2461 y Fq(C)784 2497 y Ft(N)g Fx(b)s(e)37 b Ft(C)7 b Fx(-como)s(dules)36 b(and)i(let)f Ft(f)46 b Fx(:)36 b Ft(M)47 b Fu(\000)-60 b(!)36 b Ft(N)47 b Fx(b)s(e)38 b(a)f Fp(K)18 b Fx(-linear)41 b(map.)57 b(The)0 2613 y(map)32 b Ft(f)43 b Fx(is)32 b(called)f(a)i Fs(homomorphism)f(of)j(c)-5 b(omo)g(dules)31 b Fx(if)h(the)h(diagram)1321 3285 y Ft(N)603 b(C)29 b Fu(\012)22 b Ft(N)p 1438 3257 535 4 v 1890 3255 a Fl(-)1661 3320 y Fq(\016)1692 3331 y Fk(N)1313 2798 y Ft(M)586 b(C)29 b Fu(\012)23 b Ft(M)p 1446 2769 519 4 v 1882 2767 a Fl(-)1656 2737 y Fq(\016)1687 2748 y Fk(M)p 1364 3187 4 351 v 1365 3187 a Fl(?)1285 3028 y Fq(f)p 2143 3187 V 2145 3187 a Fl(?)2184 3028 y Fn(1)p Fr(\012)p Fq(f)0 3452 y Fx(comm)m(utes.)149 3568 y(The)28 b(left)e Ft(C)7 b Fx(-como)s(dules)26 b(and)h(their)g(homomorphisms)d(form)i(the)h Fs(c)-5 b(ate)g(gory)2983 3532 y Fq(C)3042 3568 y Fu(M)30 b Fs(of)f(c)-5 b(omo)g(d-)0 3684 y(ules)p Fx(.)149 3800 y(Let)48 b Ft(N)59 b Fx(b)s(e)47 b(an)h(arbitrary)f Fp(K)17 b Fx(-mo)s(dule)52 b(and)c Ft(M)58 b Fx(b)s(e)48 b(a)g Ft(C)7 b Fx(-como)s(dule.)87 b(Then)49 b(there)f(is)f(a)0 3917 y(bijection)31 b(b)s(et)m(w)m(een)k(all)30 b(m)m(ultilinear)f (maps)1329 4084 y Ft(f)38 b Fx(:)28 b Ft(C)h Fu(\002)23 b Ft(:)17 b(:)g(:)22 b Fu(\002)g Ft(M)39 b Fu(\000)-60 b(!)27 b Ft(N)0 4251 y Fx(and)33 b(all)d(linear)h(maps)1307 4378 y Ft(f)1366 4337 y Fr(0)1416 4378 y Fx(:)d Ft(C)h Fu(\012)23 b Ft(:)17 b(:)g(:)k Fu(\012)i Ft(M)38 b Fu(\000)-60 b(!)28 b Ft(N)5 b(:)0 4525 y Fx(These)34 b(maps)f(are)f(asso)s(ciated)h (to)f(eac)m(h)h(other)g(b)m(y)h(the)f(form)m(ula)940 4692 y Ft(f)11 b Fx(\()p Ft(c)1079 4707 y Fn(1)1118 4692 y Ft(;)17 b(:)g(:)g(:)32 b(;)17 b(c)1395 4707 y Fq(n)1442 4692 y Ft(;)g(m)p Fx(\))27 b(=)h Ft(f)1799 4651 y Fr(0)1822 4692 y Fx(\()p Ft(c)1902 4707 y Fn(1)1963 4692 y Fu(\012)23 b Ft(:)17 b(:)g(:)22 b Fu(\012)g Ft(c)2341 4707 y Fq(n)2410 4692 y Fu(\012)h Ft(m)p Fx(\))p Ft(:)0 4859 y Fx(F)-8 b(or)32 b Ft(m)c Fu(2)g Ft(M)43 b Fx(w)m(e)34 b(de\014ne)921 4932 y Ff(X)1081 5027 y Ft(f)11 b Fx(\()p Ft(m)1263 5042 y Fn(\(1\))1358 5027 y Ft(;)17 b(:)g(:)g(:)32 b(;)17 b(m)1678 5042 y Fn(\()p Fq(n)p Fn(\))1780 5027 y Ft(;)g(m)1909 5042 y Fn(\()p Fq(M)7 b Fn(\))2043 5027 y Fx(\))27 b(:=)h Ft(f)2298 4986 y Fr(0)2321 5027 y Fx(\()p Ft(\016)2406 4986 y Fq(n)2453 5027 y Fx(\()p Ft(m)p Fx(\)\))p Ft(;)0 5216 y Fx(where)34 b Ft(\016)329 5179 y Fq(n)408 5216 y Fx(denotes)g(the)f Ft(n)p Fx(-fold)e(application)f(of)i Ft(\016)t Fx(,)h(i.e.)43 b Ft(\016)2149 5179 y Fq(n)2223 5216 y Fx(=)28 b(\(1)22 b Fu(\012)g Ft(:)17 b(:)g(:)22 b Fu(\012)h Fx(1)f Fu(\012)g Ft(\016)t Fx(\))g Fu(\016)g Fx(\(1)g Fu(\012)h Ft(\016)t Fx(\))f Fu(\016)g Ft(\016)t Fx(.)p eop %%Page: 31 31 31 30 bop 1474 236 a Fo(6.)76 b(CO)n(ALGEBRAS)1371 b(31)149 425 y Fx(In)33 b(particular)e(w)m(e)j(obtain)d(for)h(the)h(bilinear)e (map)g Fu(\012)d Fx(:)g Ft(C)h Fu(\002)23 b Ft(M)38 b Fu(\000)-60 b(!)28 b Ft(C)h Fu(\012)23 b Ft(M)1276 531 y Ff(X)1437 625 y Ft(m)1522 641 y Fn(\(1\))1639 625 y Fu(\012)f Ft(m)1823 641 y Fn(\()p Fq(M)7 b Fn(\))1985 625 y Fx(=)28 b Ft(\016)t Fx(\()p Ft(m)p Fx(\))p Ft(;)0 835 y Fx(and)33 b(for)f(the)h(m)m(ultilinear)c(map)i Fu(\012)1296 799 y Fn(2)1364 835 y Fx(:)d Ft(C)h Fu(\002)22 b Ft(C)30 b Fu(\002)22 b Ft(M)39 b Fu(\000)-60 b(!)27 b Ft(C)i Fu(\012)23 b Ft(C)29 b Fu(\012)23 b Ft(M)667 941 y Ff(X)827 1035 y Ft(m)912 1051 y Fn(\(1\))1029 1035 y Fu(\012)g Ft(m)1214 1051 y Fn(\(2\))1331 1035 y Fu(\012)f Ft(m)1515 1051 y Fn(\()p Fq(M)7 b Fn(\))1677 1035 y Fx(=)28 b(\(1)22 b Fu(\012)g Ft(\016)t Fx(\))p Ft(\016)t Fx(\()p Ft(c)p Fx(\))28 b(=)f(\(\001)22 b Fu(\012)h Fx(1\))p Ft(\016)t Fx(\()p Ft(m)p Fx(\))p Ft(:)149 1241 y Fv(Problem)36 b(8.6.13.)49 b Fx(Sho)m(w)39 b(that)e(a)g(\014nite)f(dimensional)f(v)m (ector)j(space)h Ft(V)59 b Fx(is)36 b(a)h(como)s(dule)0 1358 y(o)m(v)m(er)42 b(the)g(coalgebra)e Ft(V)49 b Fu(\012)29 b Ft(V)1126 1321 y Fr(\003)1207 1358 y Fx(as)41 b(de\014ned)i(in)d (problem)g(8.11.1)h(with)f(the)i(coaction)e Ft(\016)t Fx(\()p Ft(v)t Fx(\))i(:=)0 1399 y Ff(P)122 1474 y Ft(v)26 b Fu(\012)c Ft(v)345 1438 y Fr(\003)341 1498 y Fq(i)407 1474 y Fu(\012)g Ft(v)553 1489 y Fq(i)609 1474 y Fu(2)28 b Fx(\()p Ft(V)44 b Fu(\012)23 b Ft(V)1020 1438 y Fr(\003)1060 1474 y Fx(\))f Fu(\012)g Ft(V)54 b Fx(where)1612 1399 y Ff(P)1734 1474 y Ft(v)1785 1438 y Fr(\003)1781 1498 y Fq(i)1846 1474 y Fu(\012)23 b Ft(v)1993 1489 y Fq(i)2054 1474 y Fx(is)32 b(the)h(dual)f(basis)g(of)g Ft(V)54 b Fx(in)32 b Ft(V)3191 1438 y Fr(\003)3252 1474 y Fu(\012)23 b Ft(V)e Fx(.)149 1664 y Fv(Theorem)38 b(8.6.11.)49 b Fs(\(F)-7 b(undamental)35 b(The)-5 b(or)g(em)36 b(for)i(Como)-5 b(dules\))36 b(L)-5 b(et)37 b Fp(K)61 b Fs(b)-5 b(e)37 b(a)g(\014eld.)52 b(L)-5 b(et)0 1780 y Ft(M)36 b Fs(b)-5 b(e)25 b(a)g(left)g Ft(C)7 b Fs(-c)-5 b(omo)g(dule)24 b(and)h(let)h Ft(m)i Fu(2)g Ft(M)36 b Fs(b)-5 b(e)25 b(given.)40 b(Then)25 b(ther)-5 b(e)25 b(exists)g(a)g(\014nite)g (dimensional)0 1896 y(sub)-5 b(c)g(o)g(algebr)g(a)40 b Ft(C)630 1860 y Fr(0)694 1896 y Fu(\022)h Ft(C)48 b Fs(and)41 b(a)h(\014nite)f(dimensional)f Ft(C)2103 1860 y Fr(0)2126 1896 y Fs(-c)-5 b(omo)g(dule)41 b Ft(M)2693 1860 y Fr(0)2758 1896 y Fs(with)h Ft(m)f Fu(2)f Ft(M)3313 1860 y Fr(0)3377 1896 y Fu(\022)h Ft(M)0 2012 y Fs(wher)-5 b(e)34 b Ft(M)379 1976 y Fr(0)431 2012 y Fu(\022)28 b Ft(M)46 b Fs(is)34 b(a)h Fp(K)18 b Fs(-submo)-5 b(dule,)40 b(such)34 b(that)h(the)g(diagr)-5 b(am)1302 2696 y Ft(M)587 b(C)29 b Fu(\012)22 b Ft(M)p 1435 2668 519 4 v 1871 2666 a Fl(-)1678 2731 y Fq(\016)1290 2211 y Ft(M)1394 2174 y Fr(0)1959 2211 y Ft(C)2036 2174 y Fr(0)2082 2211 y Fu(\012)g Ft(M)2285 2174 y Fr(0)p 1447 2180 484 4 v 1848 2178 a Fl(-)1661 2159 y Fq(\016)1694 2135 y Fh(0)p 1353 2597 4 351 v 1354 2597 a Fl(?)p 2133 2597 V 697 w(?)0 2867 y Fs(c)-5 b(ommutes.)149 3057 y Fv(Corollary)36 b(8.6.12.)50 b Fs(1.)45 b(Each)35 b(element)f Ft(c)28 b Fu(2)h Ft(C)42 b Fs(of)35 b(a)g(c)-5 b(o)g(algebr)g(a)34 b(is)g(c)-5 b(ontaine)g(d)35 b(in)f(a)h(\014nite)0 3174 y(dimensional)e(sub)-5 b(c)g(o)g(algebr)g(a)34 b(of)g Ft(C)7 b Fs(.)149 3290 y(2.)87 b(Each)48 b(element)h Ft(m)54 b Fu(2)g Ft(M)59 b Fs(of)49 b(a)g(c)-5 b(omo)g(dule)48 b(is)g(c)-5 b(ontaine)g(d)48 b(in)h(a)f(\014nite)h(dimensional)0 3406 y(sub)-5 b(c)g(omo)g(dule)34 b(of)h Ft(M)10 b Fs(.)149 3596 y Fv(Corollary)36 b(8.6.13.)50 b Fs(1.)75 b(Each)44 b(\014nite)h(dimensional)e(subsp)-5 b(ac)g(e)44 b Ft(V)67 b Fs(of)45 b(a)f(c)-5 b(o)g(algebr)g(a)44 b Ft(C)52 b Fs(is)0 3712 y(c)-5 b(ontaine)g(d)34 b(in)g(a)h(\014nite)g(dimensional) e(sub)-5 b(c)g(o)g(algebr)g(a)33 b Ft(C)2058 3676 y Fr(0)2116 3712 y Fs(of)i Ft(C)7 b Fs(.)149 3828 y(2.)56 b(Each)38 b(\014nite)h(dimensional)e(subsp)-5 b(ac)g(e)37 b Ft(V)61 b Fs(of)38 b(a)g(c)-5 b(omo)g(dule)38 b Ft(M)49 b Fs(is)39 b(c)-5 b(ontaine)g(d)38 b(in)g(a)g(\014nite)0 3945 y(dimensional)33 b(sub)-5 b(c)g(omo)g(dule)34 b Ft(M)1200 3908 y Fr(0)1259 3945 y Fs(of)g Ft(M)10 b Fs(.)149 4134 y Fv(Corollary)36 b(8.6.14.)50 b Fs(1.)d(Each)36 b(c)-5 b(o)g(algebr)g(a)34 b(is)i(a)g(union)f(of)h(\014nite)f(dimensional)f(sub)-5 b(c)g(o)g(alge-)0 4251 y(br)g(as.)149 4367 y(2.)45 b(Each)34 b(c)-5 b(omo)g(dule)34 b(is)h(a)g(union)f(of)h(\014nite)f(dimensional)f (sub)-5 b(c)g(omo)g(dules.)149 4557 y Fi(Pr)n(oof.)42 b Fx(\()p Fs(of)34 b(the)g(The)-5 b(or)g(em)p Fx(\))30 b(W)-8 b(e)32 b(can)f(assume)h(that)f Ft(m)d Fu(6)p Fx(=)g(0)j(for)g (else)g(w)m(e)i(can)f(use)g Ft(M)3396 4521 y Fr(0)3448 4557 y Fx(=)27 b(0)0 4673 y(and)33 b Ft(C)267 4637 y Fr(0)318 4673 y Fx(=)27 b(0.)149 4789 y(Under)f(the)f(represen)m (tations)h(of)e Ft(\016)t Fx(\()p Ft(m)p Fx(\))k Fu(2)g Ft(C)13 b Fu(\012)6 b Ft(M)36 b Fx(as)25 b(\014nite)f(sums)h(of)g (decomp)s(osable)f(tensors)0 4906 y(pic)m(k)33 b(one)1397 5124 y Ft(\016)t Fx(\()p Ft(m)p Fx(\))28 b(=)1792 4999 y Fq(s)1737 5029 y Ff(X)1752 5239 y Fq(i)p Fn(=1)1897 5124 y Ft(c)1939 5139 y Fq(i)1990 5124 y Fu(\012)22 b Ft(m)2174 5139 y Fq(i)p eop %%Page: 32 32 32 31 bop 0 236 a Fo(32)1456 b(8.)77 b(TOOLBO)n(X)0 425 y Fx(of)41 b(shortest)h(length)f Ft(s)p Fx(.)69 b(Then)42 b(the)g(families)c(\()p Ft(c)1831 440 y Fq(i)1859 425 y Fu(j)p Ft(i)43 b Fx(=)f(1)p Ft(;)17 b(:)g(:)g(:)32 b(;)17 b(s)p Fx(\))41 b(and)g(\()p Ft(m)2811 440 y Fq(i)2839 425 y Fu(j)p Ft(i)i Fx(=)f(1)p Ft(;)17 b(:)g(:)g(:)32 b(;)17 b(s)p Fx(\))41 b(are)0 541 y(linearly)31 b(indep)s(enden)m(t.)44 b(Cho)s(ose)34 b(co)s(e\016cien)m(ts)f Ft(c)1815 556 y Fq(ij)1903 541 y Fu(2)c Ft(C)39 b Fx(suc)m(h)34 b(that)1024 807 y(\001\()p Ft(c)1185 822 y Fq(j)1221 807 y Fx(\))28 b(=)1450 682 y Fq(t)1391 712 y Ff(X)1405 922 y Fq(i)p Fn(=1)1551 807 y Ft(c)1593 822 y Fq(i)1643 807 y Fu(\012)23 b Ft(c)1785 822 y Fq(ij)1846 807 y Ft(;)114 b Fu(8)p Ft(j)34 b Fx(=)27 b(1)p Ft(;)17 b(:)g(:)g(:)33 b(;)17 b(s;)0 1069 y Fx(b)m(y)38 b(suitably)e(extending)i(the)f(linearly)e (indep)s(enden)m(t)j(family)d(\()p Ft(c)2434 1084 y Fq(i)2462 1069 y Fu(j)p Ft(i)g Fx(=)g(1)p Ft(;)17 b(:)g(:)g(:)32 b(;)17 b(s)p Fx(\))37 b(to)f(a)h(linearly)0 1186 y(indep)s(enden)m(t)d (family)c(\()p Ft(c)931 1201 y Fq(i)959 1186 y Fu(j)p Ft(i)e Fx(=)f(1)p Ft(;)17 b(:)g(:)g(:)32 b(;)17 b(t)p Fx(\))33 b(and)f Ft(t)c Fu(\025)g Ft(s)p Fx(.)149 1302 y(W)-8 b(e)41 b(\014rst)f(sho)m(w)h(that)e(w)m(e)i(can)f(c)m(ho)s(ose)h Ft(t)f Fx(=)g Ft(s)p Fx(.)65 b(By)41 b(coasso)s(ciativit)m(y)d(w)m(e)j (ha)m(v)m(e)3185 1227 y Ff(P)3290 1253 y Fq(s)3290 1331 y(i)p Fn(=1)3425 1302 y Ft(c)3467 1317 y Fq(i)3523 1302 y Fu(\012)0 1429 y Ft(\016)t Fx(\()p Ft(m)170 1444 y Fq(i)198 1429 y Fx(\))c(=)386 1354 y Ff(P)491 1380 y Fq(s)491 1458 y(j)t Fn(=1)635 1429 y Fx(\001\()p Ft(c)796 1444 y Fq(j)832 1429 y Fx(\))26 b Fu(\012)h Ft(m)1085 1444 y Fq(j)1158 1429 y Fx(=)1271 1354 y Ff(P)1376 1380 y Fq(s)1376 1458 y(j)t Fn(=1)1520 1354 y Ff(P)1625 1380 y Fq(t)1625 1458 y(i)p Fn(=1)1760 1429 y Ft(c)1802 1444 y Fq(i)1856 1429 y Fu(\012)f Ft(c)2001 1444 y Fq(ij)2088 1429 y Fu(\012)g Ft(m)2276 1444 y Fq(j)2313 1429 y Fx(.)60 b(Since)38 b(the)g Ft(c)2875 1444 y Fq(i)2941 1429 y Fx(and)h(the)f Ft(m)3395 1444 y Fq(j)3470 1429 y Fx(are)0 1548 y(linearly)31 b(indep)s(enden)m(t)i(w)m(e)h(can)f(compare)f(co)s (e\016cien)m(ts)i(and)e(get)1017 1793 y Ft(\016)t Fx(\()p Ft(m)1187 1808 y Fq(i)1216 1793 y Fx(\))27 b(=)1441 1668 y Fq(s)1385 1698 y Ff(X)1396 1908 y Fq(j)t Fn(=1)1545 1793 y Ft(c)1587 1808 y Fq(ij)1670 1793 y Fu(\012)c Ft(m)1855 1808 y Fq(j)1892 1793 y Ft(;)114 b Fu(8)p Ft(i)28 b Fx(=)g(1)p Ft(;)17 b(:)g(:)g(:)32 b(;)17 b(s)-2583 b Fx(\(1\))0 2070 y(and)33 b(0)27 b(=)370 1996 y Ff(P)475 2022 y Fq(s)475 2099 y(j)t Fn(=1)618 2070 y Ft(c)660 2085 y Fq(ij)743 2070 y Fu(\012)22 b Ft(m)927 2085 y Fq(j)997 2070 y Fx(for)32 b Ft(i)c(>)f(s)p Fx(.)44 b(The)33 b(last)f(statemen)m(t)h(implies)1166 2244 y Ft(c)1208 2259 y Fq(ij)1297 2244 y Fx(=)27 b(0)p Ft(;)114 b Fu(8)p Ft(i)28 b(>)g(s;)17 b(j)33 b Fx(=)28 b(1)p Ft(;)17 b(:)g(:)g(:)32 b(;)17 b(s:)0 2414 y Fx(Hence)34 b(w)m(e)g(get)e Ft(t)c Fx(=)g Ft(s)k Fx(and)1024 2659 y(\001\()p Ft(c)1185 2674 y Fq(j)1221 2659 y Fx(\))c(=)1446 2535 y Fq(s)1391 2565 y Ff(X)1405 2774 y Fq(i)p Fn(=1)1551 2659 y Ft(c)1593 2674 y Fq(i)1643 2659 y Fu(\012)23 b Ft(c)1785 2674 y Fq(ij)1846 2659 y Ft(;)114 b Fu(8)p Ft(j)34 b Fx(=)27 b(1)p Ft(;)17 b(:)g(:)g(:)33 b(;)17 b(s:)149 2922 y Fx(De\014ne)46 b(\014nite)f(dimensional)e(subspaces)48 b Ft(C)1812 2885 y Fr(0)1885 2922 y Fx(=)h Fu(h)p Ft(c)2091 2937 y Fq(ij)2151 2922 y Fu(j)p Ft(i;)17 b(j)56 b Fx(=)49 b(1)p Ft(;)17 b(:)g(:)g(:)32 b(;)17 b(s)p Fu(i)49 b(\022)h Ft(C)i Fx(and)46 b Ft(M)3451 2885 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y(\001\()p Ft(c)1498 4095 y Fq(ij)1559 4080 y Fx(\))27 b(=)1784 3956 y Fq(s)1728 3986 y Ff(X)1736 4198 y Fq(k)r Fn(=1)1888 4080 y Ft(c)1930 4095 y Fq(ik)2019 4080 y Fu(\012)c Ft(c)2161 4095 y Fq(k)r(j)2236 4080 y Ft(:)-2263 b Fx(\(2\))p 3529 4336 4 66 v 3533 4274 59 4 v 3533 4336 V 3591 4336 4 66 v 149 4518 a Fv(Remark)37 b(8.6.15.)50 b Fx(W)-8 b(e)42 b(giv)m(e)h(a)f(sk)m(etc)m(h)i(of)e(a)g(second)i(pro)s (of)d(whic)m(h)i(is)f(somewhat)h(more)0 4634 y(tec)m(hnical.)d(Since)22 b Ft(C)29 b Fx(is)22 b(a)f Fp(K)d Fx(-coalgebra,)29 b(the)22 b(dual)g Ft(C)1949 4598 y Fr(\003)2010 4634 y Fx(is)g(an)g(algebra.)39 b(The)23 b(como)s(dule)e(structure)0 4751 y Ft(\016)42 b Fx(:)37 b Ft(M)49 b Fu(\000)-60 b(!)38 b Ft(C)33 b Fu(\012)26 b Ft(M)49 b Fx(leads)39 b(to)f(a)g(mo)s(dule)f(structure)j (b)m(y)f Ft(\032)f Fx(=)g(\(ev)18 b Fu(\012)p Fx(1\)\(1)26 b Fu(\012)h Ft(\016)t Fx(\))38 b(:)f Ft(C)3171 4714 y Fr(\003)3237 4751 y Fu(\012)26 b Ft(M)49 b Fu(\000)-60 b(!)0 4867 y Ft(C)77 4831 y Fr(\003)122 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b(But)43 b(it)f(do)s(es)h (not)f(dep)s(end)i(on)0 5216 y(the)h(c)m(hoice)g(of)g(the)g Ft(m)871 5231 y Fq(i)900 5216 y Fx(.)80 b(F)-8 b(urthermore)44 b(if)g(w)m(e)i(tak)m(e)f Ft(\016)t Fx(\()p Ft(m)p Fx(\))k(=)2446 5141 y Ff(P)2568 5216 y Ft(c)2610 5231 y Fq(i)2669 5216 y Fu(\012)31 b Ft(m)2862 5231 y Fq(i)2935 5216 y Fx(with)44 b(a)h(shortest)p eop %%Page: 33 33 33 32 bop 1474 236 a Fo(6.)76 b(CO)n(ALGEBRAS)1371 b(33)0 425 y Fx(represen)m(tation)47 b(then)g(the)g Ft(m)1154 440 y Fq(i)1228 425 y Fx(are)g(in)e Ft(C)1609 389 y Fr(\003)1649 425 y Ft(m)h Fx(since)h Ft(c)2075 389 y Fr(\003)2114 425 y Ft(m)52 b Fx(=)2378 350 y Ff(P)2483 425 y Fu(h)p Ft(c)2564 389 y Fr(\003)2603 425 y Ft(;)17 b(c)2689 440 y Fq(i)2717 425 y Fu(i)p Ft(m)2841 440 y Fq(i)2920 425 y Fx(=)51 b Ft(m)3132 440 y Fq(i)3207 425 y Fx(for)45 b Ft(c)3411 389 y Fr(\003)3497 425 y Fx(an)0 541 y(elemen)m(t)32 b(of)g(a)h(dual)f(basis)g(of)g(the)h Ft(c)1330 556 y Fq(i)1358 541 y Fx(.)149 658 y Ft(N)60 b Fx(is)48 b(a)h Ft(C)7 b Fx(-como)s(dule)47 b(since)i Ft(\016)t Fx(\()p Ft(c)1443 622 y Fr(\003)1482 658 y Ft(m)p Fx(\))56 b(=)1792 583 y Ff(P)1897 658 y Fu(h)p Ft(c)1978 622 y Fr(\003)2017 658 y Ft(;)17 b(c)2103 673 y Fq(i)2131 658 y Fu(i)p Ft(\016)t Fx(\()p Ft(m)2340 673 y Fq(i)2368 658 y Fx(\))55 b(=)2592 583 y Ff(P)2698 658 y Fu(h)p Ft(c)2779 622 y Fr(\003)2818 658 y Ft(;)17 b(c)2904 673 y Fq(i)p Fn(\(1\))3022 658 y Fu(i)p Ft(c)3103 673 y Fq(i)p Fn(\(2\))3254 658 y Fu(\012)34 b Ft(m)3450 673 y Fq(i)3534 658 y Fu(2)0 774 y Ft(C)29 b Fu(\012)23 b Ft(C)276 738 y Fr(\003)315 774 y Ft(m)p Fx(.)149 890 y(No)m(w)34 b(w)m(e)h(construct)f(a)g(sub)s(coalgebra)f Ft(D)j Fx(of)d Ft(C)40 b Fx(suc)m(h)35 b(that)e Ft(N)44 b Fx(is)33 b(a)g Ft(D)s Fx(-como)s(dule)e(with)j(the)0 1006 y(induced)39 b(coaction.)59 b(Let)38 b Ft(D)i Fx(:=)d Ft(N)g Fu(\012)26 b Ft(N)1555 970 y Fr(\003)1595 1006 y Fx(.)60 b(By)39 b(8.13)e Ft(N)49 b Fx(is)38 b(a)f(como)s(dule)g(o)m (v)m(er)i(the)g(coalgebra)0 1123 y Ft(N)g Fu(\012)29 b Ft(N)311 1086 y Fr(\003)351 1123 y Fx(.)71 b(Construct)43 b(a)e(linear)f(map)h Ft(\036)i Fx(:)g Ft(D)j Fu(\000)-60 b(!)44 b Ft(C)k Fx(b)m(y)43 b Ft(n)28 b Fu(\012)h Ft(n)2482 1086 y Fr(\003)2565 1123 y Fu(7!)2708 1048 y Ff(P)2830 1123 y Ft(n)2888 1138 y Fn(\(1\))2982 1123 y Fu(h)p Ft(n)3079 1086 y Fr(\003)3119 1123 y Ft(;)17 b(n)3221 1138 y Fn(\()p Fq(N)7 b Fn(\))3343 1123 y Fu(i)p Fx(.)70 b(By)0 1239 y(de\014nition)32 b(of)g(the)h(dual)e(basis)i(w)m(e)h(ha)m(v)m(e)f Ft(n)28 b Fx(=)1727 1164 y Ff(P)1849 1239 y Ft(n)1907 1254 y Fq(i)1935 1239 y Fu(h)p Ft(n)2032 1203 y Fr(\003)2032 1263 y Fq(i)2072 1239 y Ft(;)17 b(n)p Fu(i)p Fx(.)43 b(Th)m(us)34 b(w)m(e)f(get)342 1394 y(\()p Ft(\036)21 b Fu(\012)i Ft(\036)p Fx(\)\001)736 1409 y Fq(D)800 1394 y Fx(\()p Ft(n)f Fu(\012)h Ft(n)1076 1358 y Fr(\003)1116 1394 y Fx(\))32 b(=)c(\()p Ft(\036)21 b Fu(\012)i Ft(\036)p Fx(\)\()1641 1319 y Ff(P)1762 1394 y Ft(n)g Fu(\012)f Ft(n)2000 1358 y Fr(\003)2000 1419 y Fq(i)2062 1394 y Fu(\012)h Ft(n)2220 1409 y Fq(i)2270 1394 y Fu(\012)g Ft(n)2428 1358 y Fr(\003)2468 1394 y Fx(\))1186 1510 y(=)1290 1436 y Ff(P)1411 1510 y Ft(n)1469 1526 y Fn(\(1\))1564 1510 y Fu(h)p Ft(n)1661 1474 y Fr(\003)1661 1535 y Fq(i)1700 1510 y Ft(;)17 b(n)1802 1526 y Fn(\()p Fq(N)7 b Fn(\))1924 1510 y Fu(i)22 b(\012)h Ft(n)2143 1526 y Fq(i)p Fn(\(1\))2261 1510 y Fu(h)p Ft(n)2358 1474 y Fr(\003)2398 1510 y Ft(;)17 b(n)2500 1526 y Fq(i)p Fn(\()p Fq(N)7 b Fn(\))2646 1510 y Fu(i)1186 1627 y Fx(=)1290 1552 y Ff(P)1411 1627 y Ft(n)1469 1642 y Fn(\(1\))1586 1627 y Fu(\012)23 b Ft(n)1744 1642 y Fq(i)p Fn(\(1\))1862 1627 y Fu(h)p Ft(n)1959 1590 y Fr(\003)1999 1627 y Ft(;)17 b(n)2101 1642 y Fq(i)p Fn(\()p Fq(N)7 b Fn(\))2247 1627 y Fu(ih)p Ft(n)2383 1590 y Fr(\003)2383 1651 y Fq(i)2422 1627 y Ft(;)17 b(n)2524 1642 y Fn(\()p Fq(N)7 b Fn(\))2646 1627 y Fu(i)1186 1743 y Fx(=)1290 1668 y Ff(P)1411 1743 y Ft(n)1469 1758 y Fn(\(1\))1586 1743 y Fu(\012)23 b Ft(n)1744 1758 y Fn(\(2\))1838 1743 y Fu(h)p Ft(n)1935 1707 y Fr(\003)1975 1743 y Ft(;)17 b(n)2077 1758 y Fn(\()p Fq(N)7 b Fn(\))2199 1743 y Fu(i)27 b Fx(=)2369 1668 y Ff(P)2490 1743 y Fx(\001)2571 1758 y Fq(C)2631 1743 y Fx(\()p Ft(n)2727 1758 y Fn(\(1\))2821 1743 y Fx(\))p Fu(h)p Ft(n)2956 1707 y Fr(\003)2996 1743 y Ft(;)17 b(n)3098 1758 y Fn(\()p Fq(N)7 b Fn(\))3220 1743 y Fu(i)1186 1859 y Fx(=)28 b(\001)1371 1874 y Fq(C)1430 1859 y Ft(\036)p Fx(\()p Ft(n)22 b Fu(\012)h Ft(n)1764 1823 y Fr(\003)1803 1859 y Fx(\))p Ft(:)0 2018 y Fx(F)-8 b(urthermore)45 b Ft(")624 2033 y Fq(C)683 2018 y Ft(\036)p Fx(\()p Ft(n)31 b Fu(\012)h Ft(n)1035 1981 y Fr(\003)1075 2018 y Fx(\))50 b(=)h Ft(")p Fx(\()1374 1943 y Ff(P)1495 2018 y Ft(n)1553 2033 y Fn(\(1\))1648 2018 y Fu(h)p Ft(n)1745 1981 y Fr(\003)1784 2018 y Ft(;)17 b(n)1886 2033 y Fn(\()p Fq(N)7 b Fn(\))2008 2018 y Fu(i)51 b Fx(=)f Fu(h)p Ft(n)2321 1981 y Fr(\003)2361 2018 y Ft(;)2405 1943 y Ff(P)2526 2018 y Ft(")p Fx(\()p Ft(n)2668 2033 y Fn(\(1\))2762 2018 y Fx(\))p Ft(n)2858 2033 y Fn(\()p Fq(N)7 b Fn(\))2981 2018 y Fu(i)50 b Fx(=)g Fu(h)p Ft(n)3293 1981 y Fr(\003)3333 2018 y Ft(;)17 b(n)p Fu(i)50 b Fx(=)0 2134 y Ft(")p Fx(\()p Ft(n)6 b Fu(\012)g Ft(n)289 2098 y Fr(\003)329 2134 y Fx(\).)41 b(Hence)26 b Ft(\036)h Fx(:)h Ft(D)i Fu(\000)-60 b(!)28 b Ft(C)k Fx(is)24 b(a)g(homomorphism)e(of)j(coalgebras,)g Ft(D)j Fx(is)c(\014nite)g(dimensional)0 2250 y(and)39 b(the)h(image)d Ft(C)736 2214 y Fr(0)798 2250 y Fx(:=)i Ft(\036)p Fx(\()p Ft(D)s Fx(\))f(is)h(a)f(\014nite)h(dimensional)e(sub) s(coalgebra)h(of)h Ft(C)7 b Fx(.)63 b(Clearly)38 b Ft(N)49 b Fx(is)0 2366 y(also)32 b(a)g Ft(C)354 2330 y Fr(0)377 2366 y Fx(-como)s(dule,)f(since)i(it)f(is)g(a)g Ft(D)s Fx(-como)s(dule.)149 2482 y(Finally)d(w)m(e)k(sho)m(w)f(that)g(the)f Ft(D)s Fx(-como)s(dule)f(structure)i(on)g Ft(N)42 b Fx(if)30 b(lifted)g(to)h(the)h Ft(C)7 b Fx(-como)s(dule)0 2599 y(structure)34 b(coincides)e(with)g(the)h(one)g(de\014ned)h(on)f Ft(M)10 b Fx(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)126 2729 y Ft(\016)169 2744 y Fq(C)229 2729 y Fx(\()p Ft(c)309 2693 y Fr(\003)348 2729 y Ft(m)p Fx(\))g(=)27 b Ft(\016)650 2744 y Fq(C)710 2729 y Fx(\()748 2654 y Ff(P)853 2729 y Fu(h)p Ft(c)934 2693 y Fr(\003)973 2729 y Ft(;)17 b(m)1102 2745 y Fn(\(1\))1196 2729 y Fu(i)p Ft(m)1320 2745 y Fn(\()p Fq(M)7 b Fn(\))1454 2729 y Fx(\))28 b(=)1623 2654 y Ff(P)1729 2729 y Fu(h)p Ft(c)1810 2693 y Fr(\003)1849 2729 y Ft(;)17 b(m)1978 2745 y Fn(\(1\))2072 2729 y Fu(i)p Ft(m)2196 2745 y Fn(\(2\))2313 2729 y Fu(\012)22 b Ft(m)2497 2745 y Fn(\()p Fq(M)7 b Fn(\))504 2845 y Fx(=)607 2771 y Ff(P)713 2845 y Fu(h)p Ft(c)794 2809 y Fr(\003)833 2845 y Ft(;)17 b(m)962 2861 y Fn(\(1\))1056 2845 y Fu(i)p Ft(m)1180 2861 y Fn(\(2\))1297 2845 y Fu(\012)22 b Ft(m)1481 2860 y Fq(i)1510 2845 y Fu(h)p Ft(m)1634 2809 y Fr(\003)1634 2870 y Fq(i)1673 2845 y Ft(;)17 b(m)1802 2861 y Fn(\()p Fq(M)7 b Fn(\))1936 2845 y Fu(i)28 b Fx(=)2106 2771 y Ff(P)2211 2845 y Fu(h)p Ft(c)2292 2809 y Fr(\003)2331 2845 y Ft(;)17 b(m)2460 2861 y Fn(\(1\))2555 2845 y Fu(i)p Ft(m)2679 2861 y Fn(\(2\))2773 2845 y Fu(h)p Ft(m)2897 2809 y Fr(\003)2897 2870 y Fq(i)2937 2845 y Ft(;)g(m)3066 2861 y Fn(\()p Fq(M)7 b Fn(\))3200 2845 y Fu(i)21 b(\012)i Ft(m)3445 2860 y Fq(i)504 2962 y Fx(=)k(\()p Ft(\036)22 b Fu(\012)h Fx(1\)\()950 2887 y Ff(P)1055 2962 y Fu(h)p Ft(c)1136 2925 y Fr(\003)1175 2962 y Ft(;)17 b(m)1304 2977 y Fn(\(1\))1398 2962 y Fu(i)p Ft(m)1522 2977 y Fn(\()p Fq(M)7 b Fn(\))1678 2962 y Fu(\012)23 b Ft(m)1863 2925 y Fr(\003)1863 2986 y Fq(i)1925 2962 y Fu(\012)f Ft(m)2109 2977 y Fq(i)2138 2962 y Fx(\))28 b(=)f(\()p Ft(\036)22 b Fu(\012)h Fx(1\)\()2650 2887 y Ff(P)2771 2962 y Ft(c)2813 2925 y Fr(\003)2852 2962 y Ft(m)g Fu(\012)f Ft(m)3144 2925 y Fr(\003)3144 2986 y Fq(i)3206 2962 y Fu(\012)h Ft(m)3391 2977 y Fq(i)3419 2962 y Fx(\))504 3078 y(=)k(\()p Ft(\036)22 b Fu(\012)h Fx(1\))p Ft(\016)955 3093 y Fq(D)1019 3078 y Fx(\()p Ft(c)1099 3042 y Fr(\003)1138 3078 y Ft(m)p Fx(\))p Ft(:)p eop %%Page: 34 34 34 33 bop 0 236 a Fo(34)1456 b(8.)77 b(TOOLBO)n(X)1455 425 y Fv(7.)93 b(Bialgebras)149 600 y(De\014nition)37 b(8.7.1.)49 b Fx(1.)57 b(A)38 b Fs(bialgebr)-5 b(a)36 b Fx(\()p Ft(B)5 b(;)17 b Fu(r)p Ft(;)g(\021)t(;)g Fx(\001)p Ft(;)g(\017)p Fx(\))36 b(consists)j(of)d(an)i(algebra)e(\()p Ft(B)5 b(;)17 b Fu(r)p Ft(;)g(\021)t Fx(\))0 716 y(and)33 b(a)f(coalgebra)g(\()p Ft(B)5 b(;)17 b Fx(\001)p Ft(;)g(\017)p Fx(\))32 b(suc)m(h)i(that)f(the)g(diagrams)585 902 y Ft(B)27 b Fu(\012)c Ft(B)694 b(B)27 b Fu(\012)c Ft(B)k Fu(\012)22 b Ft(B)28 b Fu(\012)22 b Ft(B)p 893 874 632 4 v 1442 872 a Fl(-)1123 847 y Fn(\001)p Fr(\012)p Fn(\001)2347 1085 y(1)p Fr(\012)p Fq(\034)8 b Fr(\012)p Fn(1)2031 982 y Fl(H)2114 1024 y(H)2197 1065 y(H)2280 1107 y(H)2363 1148 y(H)2446 1190 y(H)2455 1194 y(H)-83 b(j)p 723 1584 4 644 v 725 1584 a(?)627 1285 y Fr(r)685 1682 y Ft(B)1776 b(B)27 b Fu(\012)22 b Ft(B)p 793 1654 1713 4 v 2423 1652 a Fl(-)1619 1633 y Fn(\001)2334 1292 y Ft(B)27 b Fu(\012)c Ft(B)k Fu(\012)22 b Ft(B)28 b Fu(\012)22 b Ft(B)p 2673 1584 4 254 v 2675 1584 a Fl(?)2714 1477 y Fr(r\012r)0 1822 y Fx(and)787 1941 y Fp(K)542 2428 y Ft(B)606 2158 y Fq(\021)750 2058 y Fl(\001)708 2141 y(\001)667 2224 y(\001)625 2307 y(\001)616 2326 y(\001)-42 b(\013)1007 2164 y Fq(\021)r Fr(\012)p Fq(\021)860 2058 y Fl(A)901 2141 y(A)943 2224 y(A)984 2307 y(A)994 2326 y(A)g(U)542 2424 y Ft(B)313 b(B)28 b Fu(\012)22 b Ft(B)p 650 2396 251 4 v 818 2394 a Fl(-)745 2375 y Fn(\001)1417 1937 y Ft(B)27 b Fu(\012)c Ft(B)313 b(B)p 1725 1908 V 1893 1906 a Fl(-)1821 1888 y Fr(r)1762 2429 y Fp(K)1507 2167 y Fq(\017)p Fr(\012)p Fq(\017)1591 2058 y Fl(A)1632 2141 y(A)1674 2224 y(A)1715 2307 y(A)1725 2326 y(A)-42 b(U)1982 2165 y Fq(\017)1969 2058 y Fl(\001)1927 2141 y(\001)1886 2224 y(\001)1844 2307 y(\001)1835 2326 y(\001)g(\013)2493 1941 y Fp(K)433 b(K)p 2599 1908 353 4 v 2869 1906 a Fl(-)2746 1888 y Fn(id)2736 2428 y Ft(B)2556 2158 y Fq(\021)2566 2058 y Fl(A)2607 2141 y(A)2649 2224 y(A)2690 2307 y(A)2700 2326 y(A)-42 b(U)2957 2165 y Fq(\017)2810 2326 y Fl(\001)2851 2243 y(\001)2893 2160 y(\001)2934 2077 y(\001)2944 2058 y(\001)g(\025)0 2543 y Fx(comm)m(ute,)32 b(i.e.)43 b(\001)33 b(and)f Ft(\017)h Fx(are)g(homomorphisms)d(of)i(algebras)f(resp.)45 b Fu(r)32 b Fx(and)h Ft(\021)j Fx(are)c(homomor-)0 2660 y(phisms)g(of)g(coalgebras.)149 2776 y(2.)57 b(Giv)m(en)36 b(bialgebras)g Ft(A)h Fx(and)g Ft(B)5 b Fx(.)57 b(A)37 b(map)f Ft(f)45 b Fx(:)36 b Ft(A)f Fu(\000)-60 b(!)35 b Ft(B)42 b Fx(is)36 b(called)g(a)h Fs(homomorphism)f(of)0 2892 y(bialgebr)-5 b(as)31 b Fx(if)h(it)f(is)h(a)h(homomorphism)c(of)j (algebras)g(and)h(a)f(homomorphism)e(of)i(coalgebras.)149 3008 y(3.)44 b(The)33 b(category)g(of)f(bialgebras)f(is)h(denoted)i(b)m (y)f Fp(K)18 b Fx(-)5 b Fv(Bialg)o Fx(.)149 3190 y Fv(Problem)36 b(8.7.14.)49 b Fx(1.)65 b(Let)39 b(\()p Ft(B)5 b(;)17 b Fu(r)p Ft(;)g(\021)t Fx(\))39 b(b)s(e)h(an)f(algebra)f(and)i(\()p Ft(B)5 b(;)17 b Fx(\001)p Ft(;)g(")p Fx(\))39 b(b)s(e)h(a)f(coalgebra.) 0 3306 y(The)34 b(follo)m(wing)29 b(are)k(equiv)-5 b(alen)m(t:)139 3449 y(a\))41 b(\()p Ft(B)5 b(;)17 b Fu(r)p Ft(;)g(\021)t(;)g Fx(\001)p Ft(;)g(")p Fx(\))31 b(is)h(a)g(bialgebra.)133 3565 y(b\))42 b(\001)28 b(:)g Ft(B)k Fu(\000)-60 b(!)28 b Ft(B)f Fu(\012)c Ft(B)38 b Fx(and)32 b Ft(")c Fx(:)f Ft(B)33 b Fu(\000)-60 b(!)28 b Fp(K)56 b Fx(are)33 b(homomorphisms)d (of)i Fp(K)17 b Fx(-algebras.)144 3681 y(c\))42 b Fu(r)28 b Fx(:)f Ft(B)h Fu(\012)22 b Ft(B)33 b Fu(\000)-60 b(!)28 b Ft(B)37 b Fx(and)c Ft(\021)e Fx(:)d Fp(K)51 b Fu(\000)-60 b(!)28 b Ft(B)38 b Fx(are)32 b(homomorphisms)f(of)h Fp(K)17 b Fx(-coalgebras.)149 3824 y(2.)43 b(Let)31 b Ft(B)k Fx(b)s(e)c(a)f(\014nite)h(dimensional)d(bialgebra)g(o)m(v)m(er)k (\014eld)e Fp(K)18 b Fx(.)48 b(Sho)m(w)32 b(that)e(the)h(dual)f(space)0 3940 y Ft(B)79 3904 y Fr(\003)151 3940 y Fx(is)i(a)g(bialgebra.)149 4122 y(One)j(of)e(the)i(most)e(imp)s(ortan)m(t)g(prop)s(erties)h(of)f (bialgebras)g Ft(B)39 b Fx(is)34 b(that)g(the)g(tensor)h(pro)s(duct)0 4238 y(o)m(v)m(er)f Fp(K)56 b Fx(of)32 b(t)m(w)m(o)h Ft(B)5 b Fx(-mo)s(dules)31 b(or)i(t)m(w)m(o)g Ft(B)5 b Fx(-como)s(dules)31 b(is)h(again)f(a)i Ft(B)5 b Fx(-mo)s(dule.)149 4419 y Fv(Prop)s(osition)36 b(8.7.2.)199 b Fx(1.)41 b Fs(L)-5 b(et)30 b Ft(B)36 b Fs(b)-5 b(e)29 b(a)h(bialgebr)-5 b(a.)42 b(L)-5 b(et)30 b Ft(M)41 b Fs(and)30 b Ft(N)40 b Fs(b)-5 b(e)30 b(left)g Ft(B)5 b Fs(-mo)-5 b(dules.)267 4536 y(Then)34 b Ft(M)f Fu(\012)725 4551 y Fe(K)803 4536 y Ft(N)46 b Fs(is)34 b(a)h Ft(B)5 b Fs(-mo)-5 b(dule)34 b(by)h(the)g(map)304 4728 y Ft(B)27 b Fu(\012)22 b Ft(M)33 b Fu(\012)23 b Ft(N)853 4672 y Fn(\001)p Fr(\012)p Fn(1)847 4728 y Fu(\000)-16 b(!)27 b Ft(B)h Fu(\012)22 b Ft(B)28 b Fu(\012)22 b Ft(M)33 b Fu(\012)23 b Ft(N)1780 4672 y Fn(1)p Fr(\012)p Fq(\034)8 b Fr(\012)p Fn(1)1809 4728 y Fu(\000)-16 b(!)57 b Ft(B)27 b Fu(\012)c Ft(M)33 b Fu(\012)22 b Ft(B)28 b Fu(\012)22 b Ft(N)2782 4668 y Fq(\026)p Fr(\012)p Fq(\026)2771 4728 y Fu(\000)-16 b(!)28 b Ft(M)33 b Fu(\012)22 b Ft(N)5 b(:)149 4898 y Fx(2.)42 b Fs(L)-5 b(et)38 b Ft(B)44 b Fs(b)-5 b(e)37 b(a)h(bialgebr)-5 b(a.)54 b(L)-5 b(et)38 b Ft(M)49 b Fs(and)37 b Ft(N)49 b Fs(b)-5 b(e)38 b(left)g Ft(B)5 b Fs(-c)-5 b(omo)g(dules.)53 b(Then)38 b Ft(M)d Fu(\012)3235 4913 y Fe(K)3316 4898 y Ft(N)48 b Fs(is)38 b(a)267 5014 y Ft(B)5 b Fs(-c)-5 b(omo)g(dule)34 b(by)h(the)g(map)304 5207 y Ft(M)d Fu(\012)23 b Ft(N)665 5151 y Fq(\016)r Fr(\012)p Fq(\016)646 5207 y Fu(\000)-16 b(!)28 b Ft(B)f Fu(\012)22 b Ft(M)33 b Fu(\012)23 b Ft(B)k Fu(\012)c Ft(N)1579 5151 y Fn(1)p Fr(\012)p Fq(\034)8 b Fr(\012)p Fn(1)1608 5207 y Fu(\000)-16 b(!)57 b Ft(B)27 b Fu(\012)c Ft(B)k Fu(\012)c Ft(M)33 b Fu(\012)22 b Ft(N)2576 5151 y Fr(r\012)p Fn(1)2571 5207 y Fu(\000)-17 b(!)28 b Ft(B)f Fu(\012)c Ft(M)33 b Fu(\012)22 b Ft(N)5 b(:)p eop %%Page: 35 35 35 34 bop 1489 236 a Fo(7.)77 b(BIALGEBRAS)1386 b(35)149 437 y Fx(3.)42 b Fp(K)58 b Fs(is)35 b(a)f Ft(B)5 b Fs(-mo)-5 b(dule)35 b(by)g(the)g(map)f Ft(B)27 b Fu(\012)c Fp(K)1823 410 y Fu(\030)1824 441 y Fx(=)1928 437 y Ft(B)2099 381 y Fq(")2035 437 y Fu(\000)-16 b(!)27 b Fp(K)18 b Fs(.)149 564 y Fx(4.)42 b Fp(K)58 b Fs(is)35 b(a)f Ft(B)5 b Fs(-c)-5 b(omo)g(dule)34 b(by)h(the)g(map)g Fp(K)1773 503 y Fq(\021)1712 564 y Fu(\000)-16 b(!)27 b Ft(B)2007 536 y Fu(\030)2007 568 y Fx(=)2112 564 y Ft(B)g Fu(\012)c Fp(K)17 b Fs(.)149 748 y Fi(Pr)n(oof.)42 b Fx(W)-8 b(e)33 b(giv)m(e)g(a)f(diagrammatic)d (pro)s(of)j(for)g(1.)43 b(The)34 b(asso)s(ciativit)m(y)d(la)m(w)h(is)g (giv)m(en)h(b)m(y)-1 937 y Ft(B)28 b Fu(\012)22 b Ft(B)28 b Fu(\012)22 b Ft(M)33 b Fu(\012)23 b Ft(N)363 b(B)27 b Fu(\012)c Ft(B)k Fu(\012)c Ft(B)k Fu(\012)c Ft(M)32 b Fu(\012)23 b Ft(N)p 745 909 296 4 v 958 907 a Fl(-)727 881 y Fn(1)p Fr(\012)p Fn(\001)p Fr(\012)p Fn(1)p Fr(\012)p Fn(1)2336 937 y Ft(B)28 b Fu(\012)22 b Ft(B)27 b Fu(\012)c Ft(M)33 b Fu(\012)23 b Ft(B)k Fu(\012)c Ft(N)p 2015 909 293 4 v 2225 907 a Fl(-)2007 881 y Fn(1)p Fr(\012)p Fn(1)p Fr(\012)p Fq(\034)8 b Fr(\012)p Fn(1)3512 937 y Ft(B)27 b Fu(\012)c Ft(M)33 b Fu(\012)23 b Ft(N)p 3283 909 201 4 v 3401 907 a Fl(-)3268 875 y Fn(1)p Fr(\012)p Fq(\026)p Fr(\012)p Fq(\026)481 1327 y Ft(B)k Fu(\012)c Ft(B)k Fu(\012)c Ft(B)k Fu(\012)c Ft(B)k Fu(\012)c Ft(M)32 b Fu(\012)23 b Ft(N)403 b(B)27 b Fu(\012)c Ft(B)k Fu(\012)c Ft(B)k Fu(\012)c Ft(M)33 b Fu(\012)22 b Ft(B)28 b Fu(\012)22 b Ft(N)p 1628 1299 336 4 v 1881 1297 a Fl(-)1596 1271 y Fn(1)p Fr(\012)p Fn(1)p Fr(\012)p Fn(1)p Fr(\012)p Fq(\034)8 b Fr(\012)p Fn(1)3412 1327 y Ft(B)27 b Fu(\012)c Ft(B)k Fu(\012)22 b Ft(M)33 b Fu(\012)23 b Ft(N)p 3139 1299 244 4 v 3300 1297 a Fl(-)3101 1265 y Fn(1)p Fr(\012)p Fn(1)p Fr(\012)p Fq(\026)p Fr(\012)p Fq(\026)481 1717 y Ft(B)k Fu(\012)c Ft(B)k Fu(\012)c Ft(B)k Fu(\012)c Ft(B)k Fu(\012)c Ft(M)32 b Fu(\012)23 b Ft(N)403 b(B)27 b Fu(\012)c Ft(B)k Fu(\012)c Ft(M)33 b Fu(\012)22 b Ft(B)28 b Fu(\012)22 b Ft(B)28 b Fu(\012)22 b Ft(N)p 1628 1689 336 4 v 1881 1687 a Fl(-)1482 1602 y Fn(1)p Fr(\012)p Fn(1)p Fr(\012)p Fq(\034)8 b Fn(\()p Fq(B)s Fr(\012)p Fq(B)s(;M)f Fn(\))p Fr(\012)p Fn(1)3412 1717 y Ft(B)27 b Fu(\012)c Ft(M)32 b Fu(\012)23 b Ft(B)k Fu(\012)c Ft(N)p 3139 1689 244 4 v 3300 1687 a Fl(-)3101 1655 y Fn(1)p Fr(\012)p Fq(\026)p Fr(\012)p Fn(1)p Fr(\012)p Fq(\026)100 2107 y Ft(B)k Fu(\012)c Ft(M)33 b Fu(\012)22 b Ft(N)564 b(B)28 b Fu(\012)22 b Ft(B)27 b Fu(\012)c Ft(M)33 b Fu(\012)23 b Ft(N)p 644 2079 497 4 v 1058 2077 a Fl(-)772 2051 y Fn(\001)p Fr(\012)p Fn(1)p Fr(\012)p Fn(1)2437 2107 y Ft(B)k Fu(\012)c Ft(M)33 b Fu(\012)22 b Ft(B)27 b Fu(\012)c Ft(N)p 1915 2079 494 4 v 2326 2077 a Fl(-)2052 2051 y Fn(1)p Fr(\012)p Fq(\034)8 b Fr(\012)p Fn(1)3613 2107 y Ft(M)32 b Fu(\012)23 b Ft(N)p 3182 2079 402 4 v 3501 2077 a Fl(-)3313 2045 y Fq(\026)p Fr(\012)p Fq(\026)p 356 2009 4 1034 v 358 2009 a Fl(?)397 1511 y Fr(r\012)p Fn(1)p Fr(\012)p Fn(1)p 1331 1229 4 254 v 1333 1229 a Fl(?)1372 1121 y Fn(\001)p Fr(\012)p Fn(1)p Fr(\012)p Fn(1)p Fr(\012)p Fn(1)p Fr(\012)p Fn(1)p 2696 1229 V 2698 1229 a Fl(?)2737 1121 y Fn(\001)p Fr(\012)p Fn(1)p Fr(\012)p Fn(1)p Fr(\012)p Fn(1)p Fr(\012)p Fn(1)p 3768 1229 V 3770 1229 a Fl(?)3809 1121 y Fn(\001)p Fr(\012)p Fn(1)p Fr(\012)p Fn(1)p 1331 1619 V 1333 1619 a Fl(?)1372 1510 y Fn(1)p Fr(\012)p Fq(\034)8 b Fr(\012)p Fn(1)p Fr(\012)p Fn(1)p Fr(\012)p Fn(1)p 2696 1619 V 2698 1619 a Fl(?)2737 1509 y Fn(1)p Fr(\012)p Fq(\034)g Fn(\()p Fq(B)s(;B)s Fr(\012)p Fq(M)f Fn(\))p Fr(\012)p Fn(1)p Fr(\012)p Fn(1)p 3768 1619 V 3770 1619 a Fl(?)3809 1510 y Fn(1)p Fr(\012)p Fq(\034)h Fr(\012)p Fn(1)p 1331 2009 V 1333 2009 a Fl(?)1372 1901 y Fr(r\012r\012)p Fn(1)p Fr(\012)p Fn(1)p 2696 2009 V 2698 2009 a Fl(?)2737 1901 y Fr(r\012)p Fn(1)p Fr(\012r\012)p Fn(1)p 3768 2009 V 3770 2009 a Fl(?)3809 1896 y Fq(\026)p Fr(\012)p Fq(\026)0 2251 y Fx(The)34 b(unit)d(la)m(w)i(is)f(the)h(comm)m(utativit)m(y)d(of) 699 2440 y Ft(M)j Fu(\012)22 b Ft(N)1042 2412 y Fu(\030)1042 2444 y Fx(=)1147 2440 y Fp(K)45 b Fu(\012)23 b Ft(M)33 b Fu(\012)23 b Ft(N)636 b(B)27 b Fu(\012)c Ft(M)33 b Fu(\012)22 b Ft(N)p 1690 2411 569 4 v 2176 2409 a Fl(-)1864 2378 y Fq(\021)r Fr(\012)p Fn(1)p Fr(\012)p Fn(1)1116 2830 y Fp(K)45 b Fu(\012)23 b Fp(K)46 b Fu(\012)22 b Ft(M)33 b Fu(\012)23 b Ft(N)367 b(B)28 b Fu(\012)22 b Ft(B)28 b Fu(\012)22 b Ft(M)33 b Fu(\012)23 b Ft(N)p 1858 2801 301 4 v 2076 2799 a Fl(-)1852 2768 y Fq(\021)r Fr(\012)p Fq(\021)r Fr(\012)p Fn(1)p Fr(\012)p Fn(1)697 3220 y Ft(M)33 b Fu(\012)22 b Ft(N)1039 3192 y Fu(\030)1040 3224 y Fx(=)1145 3220 y Fp(K)45 b Fu(\012)23 b Ft(M)33 b Fu(\012)22 b Fp(K)46 b Fu(\012)23 b Ft(N)338 b(B)28 b Fu(\012)22 b Ft(M)33 b Fu(\012)23 b Ft(B)k Fu(\012)c Ft(N)p 1887 3191 272 4 v 2076 3189 a Fl(-)1867 3158 y Fq(\021)r Fr(\012)p Fn(1)p Fr(\012)p Fq(\021)r Fr(\012)p Fn(1)p 1373 2731 4 254 v 1375 2731 a Fl(?)1414 2607 y Fr(\030)1414 2628 y Fn(=)p 1373 3121 V 1375 3121 a Fl(?)1414 3013 y Fn(1)p Fr(\012)p Fq(\034)8 b Fr(\012)p Fn(1)p 2543 2731 V 2545 2731 a Fl(?)2584 2624 y Fn(\001)p Fr(\012)p Fn(1)p Fr(\012)p Fn(1)p 2543 3121 V 2545 3121 a Fl(?)2584 3013 y Fn(1)p Fr(\012)p Fq(\034)g Fr(\012)p Fn(1)p 2543 3511 V 2545 3511 a Fl(?)2584 3398 y Fq(\026)p Fr(\012)p Fq(\026)p 837 3121 4 644 v 839 3121 a Fl(?)745 2812 y Fn(=)2387 3609 y Ft(M)33 b Fu(\012)23 b Ft(N)1663 3406 y Fn(1)1258 3279 y Fl(X)1341 3299 y(X)1424 3320 y(X)1507 3341 y(X)1590 3362 y(X)1673 3382 y(X)1756 3403 y(X)1839 3424 y(X)1922 3445 y(X)2005 3465 y(X)2088 3486 y(X)2171 3507 y(X)2189 3511 y(X)-83 b(z)0 3753 y Fx(The)30 b(corresp)s(onding)e (prop)s(erties)h(for)f(como)s(dules)g(follo)m(ws)f(from)h(the)h (dualized)f(diagrams.)40 b(The)0 3869 y(mo)s(dule)31 b(and)i(como)s(dule)e(prop)s(erties)i(of)f Fp(K)56 b Fx(are)32 b(easily)g(c)m(hec)m(k)m(ed.)p 3529 3869 4 66 v 3533 3807 59 4 v 3533 3869 V 3591 3869 4 66 v 149 4053 a Fv(De\014nition)37 b(8.7.3.)49 b Fx(1.)41 b(Let)27 b(\()p Ft(B)5 b(;)17 b Fu(r)p Ft(;)g(\021)t(;)g Fx(\001)p Ft(;)g(\017)p Fx(\))26 b(b)s(e)g(a)h(bialgebra.)39 b(Let)27 b Ft(A)g Fx(b)s(e)g(a)f(left)g Ft(B)5 b Fx(-mo)s(dule)0 4169 y(with)31 b(structure)i(map)e Ft(\026)c Fx(:)h Ft(B)d Fu(\012)c Ft(A)28 b Fu(\000)-60 b(!)27 b Ft(A)p Fx(.)44 b(Let)31 b(furthermore)h(\()p Ft(A;)17 b Fu(r)2531 4184 y Fq(A)2587 4169 y Ft(;)g(\021)2679 4184 y Fq(A)2736 4169 y Fx(\))32 b(b)s(e)g(an)f(algebra)f(suc)m(h)0 4286 y(that)37 b Fu(r)299 4301 y Fq(A)393 4286 y Fx(and)g Ft(\021)635 4301 y Fq(A)729 4286 y Fx(are)g(homomorphisms)d(of)j Ft(B)5 b Fx(-mo)s(dules.)55 b(Then)38 b(\()p Ft(A;)17 b Fu(r)2785 4301 y Fq(A)2842 4286 y Ft(;)g(\021)2934 4301 y Fq(A)2991 4286 y Ft(;)g(\026)p Fx(\))36 b(is)g(called)g(a)0 4402 y Ft(B)5 b Fs(-mo)-5 b(dule)34 b(algebr)-5 b(a)p Fx(.)149 4518 y(2.)41 b(Let)26 b(\()p Ft(B)5 b(;)17 b Fu(r)p Ft(;)g(\021)t(;)g Fx(\001)p Ft(;)g(\017)p Fx(\))26 b(b)s(e)g(a)f(bialgebra.)40 b(Let)26 b Ft(C)32 b Fx(b)s(e)26 b(a)g(left)f Ft(B)5 b Fx(-mo)s(dule)24 b(with)i(structure)h(map)0 4634 y Ft(\026)g Fx(:)h Ft(B)18 b Fu(\012)12 b Ft(C)35 b Fu(\000)-60 b(!)28 b Ft(C)7 b Fx(.)41 b(Let)28 b(furthermore)g(\()p Ft(C)r(;)17 b Fx(\001)1657 4649 y Fq(C)1715 4634 y Ft(;)g(")1805 4649 y Fq(C)1864 4634 y Fx(\))28 b(b)s(e)g(a)f(coalgebra)g(suc)m(h)i (that)e(\001)3065 4649 y Fq(C)3153 4634 y Fx(and)g Ft(")3383 4649 y Fq(C)3470 4634 y Fx(are)0 4751 y(homomorphisms)j(of)i Ft(B)5 b Fx(-)33 b(mo)s(dules.)42 b(Then)34 b(\()p Ft(C)r(;)17 b Fx(\001)1893 4766 y Fq(C)1951 4751 y Ft(;)g(")2041 4766 y Fq(C)2100 4751 y Ft(;)g(\026)p Fx(\))32 b(is)g(called)f(a)i Ft(B)5 b Fs(-mo)-5 b(dule)34 b(c)-5 b(o)g(algebr)g(a)p Fx(.)149 4867 y(3.)51 b(Let)36 b(\()p Ft(B)5 b(;)17 b Fu(r)p Ft(;)g(\021)t(;)g Fx(\001)p Ft(;)g(\017)p Fx(\))34 b(b)s(e)h(a)g(bialgebra.)49 b(Let)36 b Ft(A)f Fx(b)s(e)g(a)g(left)f Ft(B)5 b Fx(-como)s(dule)34 b(with)h(structure)0 4983 y(map)g Ft(\016)i Fx(:)c Ft(A)g Fu(\000)-60 b(!)33 b Ft(B)c Fu(\012)c Ft(A)p Fx(.)53 b(Let)35 b(furthermore)h(\()p Ft(A;)17 b Fu(r)1932 4998 y Fq(A)1988 4983 y Ft(;)g(\021)2080 4998 y Fq(A)2137 4983 y Fx(\))36 b(b)s(e)g(an)f(algebra)g(suc)m(h)i (that)e Fu(r)3350 4998 y Fq(A)3443 4983 y Fx(and)0 5099 y Ft(\021)48 5114 y Fq(A)139 5099 y Fx(are)g(homomorphisms)d(of)i Ft(B)5 b Fx(-como)s(dules.)48 b(Then)35 b(\()p Ft(A;)17 b Fu(r)2270 5114 y Fq(A)2327 5099 y Ft(;)g(\021)2419 5114 y Fq(A)2476 5099 y Ft(;)g(\016)t Fx(\))34 b(is)g(called)f(a)h Ft(B)5 b Fs(-c)-5 b(omo)g(dule)0 5216 y(algebr)g(a)p Fx(.)p eop %%Page: 36 36 36 35 bop 0 236 a Fo(36)1456 b(8.)77 b(TOOLBO)n(X)149 425 y Fx(4.)51 b(Let)35 b(\()p Ft(B)5 b(;)17 b Fu(r)p Ft(;)g(\021)t(;)g Fx(\001)p Ft(;)g(\017)p Fx(\))34 b(b)s(e)h(a)g (bialgebra.)48 b(Let)35 b Ft(C)42 b Fx(b)s(e)35 b(a)g(left)f Ft(B)5 b Fx(-como)s(dule)33 b(with)i(structure)0 541 y(map)e Ft(\016)g Fx(:)c Ft(C)37 b Fu(\000)-60 b(!)29 b Ft(B)f Fu(\012)23 b Ft(C)7 b Fx(.)47 b(Let)33 b(furthermore)h(\()p Ft(C)r(;)17 b Fx(\001)1907 556 y Fq(C)1965 541 y Ft(;)g(")2055 556 y Fq(C)2114 541 y Fx(\))33 b(b)s(e)h(a)f(coalgebra)g(suc)m(h)i (that)e(\001)3350 556 y Fq(C)3443 541 y Fx(and)0 658 y Ft(")46 673 y Fq(C)139 658 y Fx(are)i(homomorphisms)d(of)i Ft(B)5 b Fx(-como)s(dules.)48 b(Then)36 b(\()p Ft(C)r(;)17 b Fx(\001)2268 673 y Fq(C)2327 658 y Ft(;)g(")2417 673 y Fq(C)2475 658 y Ft(;)g(\016)t Fx(\))34 b(is)g(called)g(a)g Ft(B)5 b Fs(-c)-5 b(omo)g(dule)0 774 y(c)g(o)g(algebr)g(a)p Fx(.)149 948 y Fv(Remark)37 b(8.7.4.)50 b Fx(If)38 b(\()p Ft(C)r(;)17 b Fx(\001)1223 963 y Fq(C)1281 948 y Ft(;)g(")1371 963 y Fq(C)1430 948 y Fx(\))38 b(is)g(a)g Fp(K)17 b Fx(-coalgebra)43 b(and)c(\()p Ft(C)r(;)17 b(\026)p Fx(\))37 b(is)g(a)h Ft(B)5 b Fx(-mo)s(dule,)38 b(then)0 1064 y(\()p Ft(C)r(;)17 b Fx(\001)235 1079 y Fq(C)294 1064 y Ft(;)g(")384 1079 y Fq(C)442 1064 y Ft(;)g(\026)p Fx(\))32 b(is)g(a)g Ft(B)5 b Fx(-mo)s(dule)31 b(coalgebra)h(i\013)f Ft(\026)h Fx(is)g(a)h (homomorphism)d(of)i Fp(K)17 b Fx(-coalgebras.)149 1181 y(If)34 b(\()p Ft(A;)17 b Fu(r)486 1196 y Fq(A)543 1181 y Ft(;)g(\021)635 1196 y Fq(A)692 1181 y Fx(\))34 b(is)f(a)h Fp(K)17 b Fx(-algebra)39 b(and)34 b(\()p Ft(A;)17 b(\016)t Fx(\))33 b(is)h(a)f Ft(B)5 b Fx(-como)s(dule,)33 b(then)i(\()p Ft(A;)17 b Fu(r)3084 1196 y Fq(A)3140 1181 y Ft(;)g(\021)3232 1196 y Fq(A)3289 1181 y Ft(;)g(\016)t Fx(\))34 b(is)f(a)0 1297 y Ft(B)5 b Fx(-como)s(dule)31 b(algebra)g(i\013)h Ft(\016)k Fx(is)c(a)h(homomorphism)d(of)i Fp(K)17 b Fx(-algebras.)149 1413 y(Similar)29 b(statemen)m(t)34 b(for)e(mo)s(dule)f(algebras)h(or)g (como)s(dule)f(coalgebras)h(do)h Fs(not)f Fx(hold.)p eop %%Page: 37 37 37 36 bop 1169 236 a Fo(8.)77 b(REPRESENT)-6 b(ABLE)33 b(FUNCTORS)1065 b(37)1133 425 y Fv(8.)94 b(Represen)m(table)37 b(F)-9 b(unctors)149 600 y(De\014nition)37 b(8.8.1.)49 b Fx(Let)37 b Fu(F)44 b Fx(:)34 b Fu(C)41 b(\000)-60 b(!)34 b Fv(Set)j Fx(b)s(e)g(a)f(co)m(v)-5 b(arian)m(t)36 b(functor.)56 b(A)36 b(pair)g(\()p Ft(A;)17 b(x)p Fx(\))36 b(with)0 716 y Ft(A)g Fu(2)f(C)6 b Ft(;)17 b(x)36 b Fu(2)g(F)10 b Fx(\()p Ft(A)p Fx(\))37 b(is)f(called)g(a)h Fs(r)-5 b(epr)g(esenting)38 b(\(generic,)h(universal\))g(obje)-5 b(ct)36 b Fx(for)h Fu(F)47 b Fx(and)37 b Fu(F)46 b Fx(is)0 832 y(called)33 b(a)h Fs(r)-5 b(epr)g(esentable)35 b(functor)p Fx(,)f(if)f(for)h(eac)m(h)h Ft(B)g Fu(2)30 b(C)41 b Fx(and)34 b Ft(y)f Fu(2)e(F)10 b Fx(\()p Ft(B)5 b Fx(\))33 b(there)i(exists)g(a)e (unique)0 948 y Ft(f)38 b Fu(2)28 b Fx(Mor)357 963 y Fr(C)402 948 y Fx(\()p Ft(A;)17 b(B)5 b Fx(\))32 b(suc)m(h)i(that)e Fu(F)10 b Fx(\()p Ft(f)h Fx(\)\()p Ft(x)p Fx(\))27 b(=)h Ft(y)t Fx(:)1384 1151 y Ft(A)1381 1638 y(B)p 1419 1536 4 351 v 1421 1536 a Fl(?)1341 1377 y Fq(f)1793 1142 y Fu(F)10 b Fx(\()p Ft(A)p Fx(\))1790 1629 y Fu(F)g Fx(\()p Ft(B)5 b Fx(\))p 1907 1536 V 1908 1536 a Fl(?)1716 1377 y Fr(F)i Fn(\()p Fq(f)g Fn(\))2044 1142 y Fu(3)28 b Ft(x)2046 1622 y Fu(3)g Ft(y)p 2150 1536 V 2152 1536 a Fl(?)p 2150 1210 4 26 v 2152 1210 a(?)149 1821 y Fv(Prop)s(osition)36 b(8.8.2.)49 b Fs(L)-5 b(et)44 b Fx(\()p Ft(A;)17 b(x)p Fx(\))44 b Fs(and)f Fx(\()p Ft(B)5 b(;)17 b(y)t Fx(\))42 b Fs(b)-5 b(e)43 b(r)-5 b(epr)g(esenting)43 b(obje)-5 b(cts)43 b(for)g Fu(F)10 b Fs(.)70 b(Then)0 1937 y(ther)-5 b(e)35 b(exists)f(a)h(unique)g(isomorphism)e Ft(f)38 b Fx(:)28 b Ft(A)g Fu(\000)-58 b(!)28 b Ft(B)40 b Fs(such)35 b(that)g Fu(F)10 b Fx(\()p Ft(f)h Fx(\)\()p Ft(x)p Fx(\))27 b(=)h Ft(y)t Fs(.)1374 2140 y Ft(A)1371 2627 y(B)p 1409 2525 4 351 v 1411 2525 a Fl(?)1450 2373 y Fq(h)1374 3115 y Ft(A)p 1409 3013 V 1411 3013 a Fl(?)1450 2860 y Fq(k)1371 3602 y Ft(B)p 1409 3500 V 1411 3500 a Fl(?)1450 3348 y Fq(h)1260 2257 y Fl(\000)1226 2291 y(\000)-83 b(\011)p 1214 2886 4 585 v 1216 2886 a(?)1092 2609 y Fn(1)1127 2620 y Fk(A)1226 2988 y Fl(@)1260 3022 y(@)g(R)1211 2745 y(\000)1177 2779 y(\000)g(\011)p 1166 3373 V 1167 3373 a(?)1040 3097 y Fn(1)1075 3108 y Fk(B)1177 3476 y Fl(@)1211 3510 y(@)g(R)1783 2131 y Fu(F)10 b Fx(\()p Ft(A)p Fx(\))1780 2618 y Fu(F)g Fx(\()p Ft(B)5 b Fx(\))p 1897 2525 4 351 v 1898 2525 a Fl(?)1707 2366 y Fr(F)i Fn(\()p Fq(h)p Fn(\))1783 3106 y Fu(F)j Fx(\()p Ft(A)p Fx(\))p 1897 3013 V 1898 3013 a Fl(?)1709 2854 y Fr(F)d Fn(\()p Fq(k)r Fn(\))1780 3593 y Fu(F)j Fx(\()p Ft(B)5 b Fx(\))p 1897 3500 V 1898 3500 a Fl(?)1707 3341 y Fr(F)i Fn(\()p Fq(h)p Fn(\))1967 2257 y Fl(@)2001 2291 y(@)-83 b(R)p 2092 2886 4 585 v 2093 2886 a(?)2132 2601 y Fn(1)2167 2615 y Fh(F)5 b Fc(\()p Fk(A)p Fc(\))2001 2988 y Fl(\000)1967 3022 y(\000)-83 b(\011)2015 2745 y(@)2049 2779 y(@)g(R)p 2141 3373 V 2142 3373 a(?)2181 3089 y Fn(1)2216 3103 y Fh(F)5 b Fc(\()p Fk(B)r Fc(\))2049 3476 y Fl(\000)2015 3510 y(\000)-83 b(\011)2504 2127 y Ft(x)2507 2605 y(y)p 2531 2525 4 351 v 2532 2525 a Fl(?)p 2531 2199 4 26 v 2532 2199 a(?)p 2531 3013 4 351 v 2532 3013 a(?)p 2531 2687 4 26 v 2532 2687 a(?)2504 3102 y Ft(x)2507 3580 y(y)p 2531 3500 4 351 v 2532 3500 a Fl(?)p 2531 3174 4 26 v 2532 3174 a(?)149 3779 y Fv(Examples)37 b(8.8.3.)49 b Fx(1.)69 b(Let)41 b Ft(X)49 b Fu(2)43 b Fv(Set)d Fx(and)h(let)g Ft(R)h Fx(b)s(e)f(a)f(ring.)68 b Fu(F)51 b Fx(:)42 b Ft(R)q Fx(-)o Fv(Mo)s(d)h Fu(\000)-60 b(!)42 b Fv(Set)p Fx(,)0 3896 y Fu(F)10 b Fx(\()p Ft(M)g Fx(\))28 b(:=)g(Map\()p Ft(X)r(;)17 b(M)10 b Fx(\))33 b(is)f(a)h(co)m(v)-5 b(arian)m(t)32 b(functor.)43 b(A)33 b(represen)m(ting)h(ob)5 b(ject)33 b(for)f Fu(F)42 b Fx(is)32 b(giv)m(en)h(b)m(y)0 4012 y(\()p Ft(R)q(X)r(;)17 b(x)36 b Fx(:)f Ft(X)43 b Fu(\000)-60 b(!)35 b Ft(R)q(X)8 b Fx(\))37 b(with)f(the)i(prop)s(ert)m(y)-8 b(,)38 b(that)f(for)g(all)e(\()p Ft(M)5 b(;)17 b(y)38 b Fx(:)d Ft(X)43 b Fu(\000)-60 b(!)35 b Ft(M)10 b Fx(\))38 b(there)f(exists)h(a)0 4128 y(unique)33 b Ft(f)38 b Fu(2)29 b Fx(Hom)701 4143 y Fq(R)758 4128 y Fx(\()p Ft(R)q(X)r(;)17 b(M)10 b Fx(\))34 b(suc)m(h)g(that)e Fu(F)10 b Fx(\()p Ft(f)h Fx(\)\()p Ft(x)p Fx(\))27 b(=)h(Map\()p Ft(X)r(;)17 b(f)11 b Fx(\)\()p Ft(x)p Fx(\))28 b(=)f Ft(f)11 b(x)28 b Fx(=)g Ft(y)1493 4328 y(X)369 b(R)q(X)p 1610 4295 304 4 v 1831 4293 a Fl(-)1742 4275 y Fq(x)1683 4545 y(y)1605 4445 y Fl(@)1689 4528 y(@)1772 4611 y(@)1855 4694 y(@)1873 4713 y(@)-83 b(R)1961 4815 y Ft(M)5 b(:)p 2023 4713 4 351 v 2025 4713 a Fl(?)2064 4554 y Fq(f)0 4983 y Fx(2.)55 b(Giv)m(en)37 b(mo)s(dules)e Ft(M)898 4998 y Fq(R)993 4983 y Fx(and)1187 4998 y Fq(R)1244 4983 y Ft(N)10 b Fx(.)56 b(De\014ne)38 b Fu(F)44 b Fx(:)34 b Fv(Ab)h Fu(\000)-60 b(!)34 b Fv(Set)j Fx(b)m(y)g Fu(F)10 b Fx(\()p Ft(A)p Fx(\))34 b(:=)h(Bil)3091 4998 y Fq(R)3149 4983 y Fx(\()p Ft(M)5 b(;)17 b(N)10 b Fx(;)17 b Ft(A)p Fx(\).)0 5099 y(Then)32 b Fu(F)40 b Fx(is)31 b(a)f(co)m(v)-5 b(arian)m(t)31 b(functor.)43 b(A)31 b(represen)m(ting)h(ob)5 b(ject)32 b(for)e Fu(F)40 b Fx(is)31 b(giv)m(en)g(b)m(y)h(\()p Ft(M)d Fu(\012)3264 5114 y Fq(R)3341 5099 y Ft(N)5 b(;)17 b Fu(\012)28 b Fx(:)0 5216 y Ft(M)i Fu(\002)20 b Ft(N)38 b Fu(\000)-60 b(!)28 b Ft(M)i Fu(\012)683 5231 y Fq(R)761 5216 y Ft(N)10 b Fx(\))32 b(with)e(the)i(prop)s(ert)m(y)g(that)f(for)g (all)e(\()p Ft(A;)17 b(f)38 b Fx(:)28 b Ft(M)i Fu(\002)20 b Ft(N)38 b Fu(\000)-60 b(!)28 b Ft(A)p Fx(\))j(there)h(exists)p eop %%Page: 38 38 38 37 bop 0 236 a Fo(38)1456 b(8.)77 b(TOOLBO)n(X)0 425 y Fx(a)32 b(unique)h Ft(g)e Fu(2)d Fx(Hom\()p Ft(M)k Fu(\012)1015 440 y Fq(R)1096 425 y Ft(N)5 b(;)17 b(A)p Fx(\))32 b(suc)m(h)i(that)f Fu(F)10 b Fx(\()p Ft(g)t Fx(\)\()p Fu(\012)p Fx(\))26 b(=)i(Bil)2413 440 y Fq(R)2471 425 y Fx(\()p Ft(M)5 b(;)17 b(N)10 b Fx(;)17 b Ft(g)t Fx(\)\()p Fu(\012)p Fx(\))27 b(=)h Ft(g)t Fu(\012)f Fx(=)h Ft(f)1384 652 y(M)33 b Fu(\002)23 b Ft(N)154 b(M)33 b Fu(\012)2047 667 y Fq(R)2127 652 y Ft(N)p 1728 627 86 4 v 1731 625 a Fl(-)1743 599 y Fr(\012)1684 885 y Fq(f)1610 776 y Fl(@)1693 859 y(@)1776 942 y(@)1859 1025 y(@)1878 1044 y(@)-83 b(R)1979 1146 y Ft(A:)p 2027 1044 4 351 v 2029 1044 a Fl(?)2068 877 y Fq(g)0 1329 y Fx(3.)41 b(Giv)m(en)27 b(a)g Fp(K)18 b Fx(-mo)s(dule)31 b Ft(V)22 b Fx(.)41 b(De\014ne)28 b Fu(F)37 b Fx(:)28 b Fv(Alg)e Fu(\000)-60 b(!)28 b Fv(Set)f Fx(b)m(y)h Fu(F)10 b Fx(\()p Ft(A)p Fx(\))27 b(:=)h(Hom)o(\()p Ft(V)5 b(;)17 b(A)p Fx(\).)42 b(Then)28 b Fu(F)37 b Fx(is)26 b(a)0 1445 y(co)m(v)-5 b(arian)m(t)26 b(functor.)42 b(A)27 b(represen)m(ting)h(ob)5 b(ject)27 b(for)g Fu(F)36 b Fx(is)26 b(giv)m(en)h(b)m(y)h(\()p Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))p Ft(;)c(\023)28 b Fx(:)g Ft(V)49 b Fu(\000)-60 b(!)27 b Ft(T)14 b Fx(\()p Ft(V)22 b Fx(\)\))k(with)0 1561 y(the)39 b(prop)s(ert)m(y)g(that)f(for) g(all)e(\()p Ft(A;)17 b(f)48 b Fx(:)38 b Ft(V)59 b Fu(\000)-60 b(!)37 b Ft(A)p Fx(\))i(the)g(exists)g(a)f(unique)h Ft(g)h Fu(2)e Fx(Mor)3053 1576 y Fj(Alg)3182 1561 y Fx(\()p Ft(T)14 b Fx(\()p Ft(V)21 b Fx(\))p Ft(;)c(A)p Fx(\))0 1678 y(suc)m(h)34 b(that)e Fu(F)10 b Fx(\()p Ft(g)t Fx(\)\()p Ft(\023)p Fx(\))27 b(=)h(Hom)o(\()p Ft(V)5 b(;)17 b(g)t Fx(\)\()p Ft(\023)p Fx(\))27 b(=)h Ft(g)t(\023)f Fx(=)h Ft(f)1480 1886 y(V)358 b(T)14 b Fx(\()p Ft(V)21 b Fx(\))p 1588 1862 278 4 v 1783 1860 a Fl(-)1714 1842 y Fq(\023)1662 2121 y(f)1588 2012 y Fl(@)1671 2095 y(@)1754 2178 y(@)1837 2261 y(@)1856 2280 y(@)-83 b(R)1957 2382 y Ft(A:)p 2006 2280 4 351 v 2007 2280 a Fl(?)2046 2112 y Fq(g)0 2565 y Fx(4.)52 b(Giv)m(en)35 b(a)g Fp(K)18 b Fx(-mo)s(dule)39 b Ft(V)22 b Fx(.)52 b(De\014ne)36 b Fu(F)42 b Fx(:)33 b Fv(cAlg)e Fu(\000)-60 b(!)32 b Fv(Set)k Fx(b)m(y)g Fu(F)10 b Fx(\()p Ft(A)p Fx(\))32 b(:=)h(Hom)o(\()p Ft(V)5 b(;)17 b(A)p Fx(\).)52 b(Then)37 b Fu(F)0 2681 y Fx(is)31 b(a)h(co)m(v)-5 b(arian)m(t)32 b(functor.)43 b(A)32 b(represen)m(ting)h (ob)5 b(ject)33 b(for)e Fu(F)42 b Fx(is)31 b(giv)m(en)h(b)m(y)h(\()p Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))p Ft(;)c(\023)28 b Fx(:)g Ft(V)49 b Fu(\000)-60 b(!)28 b Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\)\))0 2797 y(with)i(the)h(prop)s(ert)m(y)g(that)f(for)g(all)f(\() p Ft(A;)17 b(f)38 b Fx(:)28 b Ft(V)49 b Fu(\000)-60 b(!)28 b Ft(A)p Fx(\))23 b(the)h(exists)g(a)f(unique)h Ft(g)31 b Fu(2)d Fx(Mor)3058 2812 y Fj(Alg)3187 2797 y Fx(\()p Ft(S)6 b Fx(\()p Ft(V)21 b Fx(\))p Ft(;)c(A)p Fx(\))0 2913 y(suc)m(h)34 b(that)e Fu(F)10 b Fx(\()p Ft(g)t Fx(\)\()p Ft(\023)p Fx(\))27 b(=)h(Hom)o(\()p Ft(V)5 b(;)17 b(g)t Fx(\)\()p Ft(\023)p Fx(\))27 b(=)h Ft(g)t(\023)f Fx(=)h Ft(f)1482 3122 y(V)360 b(S)6 b Fx(\()p Ft(V)21 b Fx(\))p 1589 3098 281 4 v 1787 3096 a Fl(-)1717 3078 y Fq(\023)1663 3357 y(f)1589 3248 y Fl(@)1672 3331 y(@)1755 3414 y(@)1838 3497 y(@)1857 3516 y(@)-83 b(R)1959 3618 y Ft(A:)p 2007 3516 4 351 v 2009 3516 a Fl(?)2048 3348 y Fq(g)149 3793 y Fv(Prop)s(osition)36 b(8.8.4.)49 b Fu(F)h Fs(has)40 b(a)h(r)-5 b(epr)g(esenting)39 b(obje)-5 b(ct)41 b Fx(\()p Ft(A;)17 b(a)p Fx(\))40 b Fs(if)h(and)f(only)g(if)g(ther)-5 b(e)41 b(is)f(a)0 3909 y(natur)-5 b(al)35 b(isomorphism)e Ft(')28 b Fx(:)f Fu(F)1165 3881 y(\030)1165 3913 y Fx(=)1270 3909 y(Mor)1446 3924 y Fr(C)1491 3909 y Fx(\()p Ft(A;)17 b Fu(\000)p Fx(\))35 b(\()p Fs(with)g Ft(a)28 b Fx(=)f Ft(')p Fx(\()p Ft(A)p Fx(\))2441 3873 y Fr(\000)p Fn(1)2535 3909 y Fx(\(1)2622 3924 y Fq(A)2679 3909 y Fx(\)\))p Fs(.)149 4117 y Fi(Pr)n(oof.)75 b Fx(=)-17 b Fu(\))60 b Fx(:)33 b(The)g(map)640 4334 y Ft(')p Fx(\()p Ft(B)5 b Fx(\))28 b(:)g Fu(F)10 b Fx(\()p Ft(B)5 b Fx(\))27 b Fu(3)h Ft(y)j Fu(7!)c Ft(f)38 b Fu(2)28 b Fx(Mor)1863 4349 y Fr(C)1908 4334 y Fx(\()p Ft(A;)17 b(B)5 b Fx(\))32 b(with)g Fu(F)10 b Fx(\()p Ft(f)h Fx(\)\()p Ft(a)p Fx(\))27 b(=)h Ft(y)0 4550 y Fx(is)k(bijectiv)m(e)h(with)f(the)h(in)m(v)m(erse)h (map)869 4767 y Ft( )t Fx(\()p Ft(B)5 b Fx(\))28 b(:)g(Mor)1350 4782 y Fr(C)1395 4767 y Fx(\()p Ft(A;)17 b(B)5 b Fx(\))27 b Fu(3)h Ft(f)39 b Fu(7!)27 b(F)10 b Fx(\()p Ft(f)h Fx(\)\()p Ft(a)p Fx(\))27 b Fu(2)h(F)10 b Fx(\()p Ft(B)5 b Fx(\))p Ft(:)0 4983 y Fx(In)35 b(fact)g(w)m(e)g(ha)m(v)m(e)h Ft(y)f Fu(7!)c Ft(f)42 b Fu(7!)31 b(F)10 b Fx(\()p Ft(f)h Fx(\)\()p Ft(a)p Fx(\))30 b(=)i Ft(y)37 b Fx(and)e Ft(f)42 b Fu(7!)31 b Ft(y)k Fx(:=)c Fu(F)10 b Fx(\()p Ft(f)h Fx(\)\()p Ft(a)p Fx(\))30 b Fu(7!)h Ft(g)k Fx(:)c Fu(F)10 b Fx(\()p Ft(g)t Fx(\)\()p Ft(a)p Fx(\))30 b(=)i Ft(y)i Fx(=)0 5099 y Fu(F)10 b Fx(\()p Ft(f)h Fx(\)\()p Ft(a)p Fx(\).)42 b(By)30 b(uniqueness)i(w)m(e)f(get)f Ft(f)38 b Fx(=)28 b Ft(g)t Fx(.)42 b(Hence)31 b(all)d Ft(')p Fx(\()p Ft(B)5 b Fx(\))29 b(are)h(bijectiv)m(e)g(with)f(in)m(v)m(erse)j (map)0 5216 y Ft( )t Fx(\()p Ft(B)5 b Fx(\).)43 b(It)33 b(is)f(su\016cien)m(t)i(to)e(sho)m(w)i(that)e Ft( )37 b Fx(is)32 b(a)g(natural)g(transformation.)p eop %%Page: 39 39 39 38 bop 1169 236 a Fo(8.)77 b(REPRESENT)-6 b(ABLE)33 b(FUNCTORS)1065 b(39)149 425 y Fx(Giv)m(en)33 b Ft(g)e Fx(:)d Ft(B)k Fu(\000)-60 b(!)28 b Ft(C)7 b Fx(.)44 b(Then)33 b(the)g(follo)m(wing)d(diagram)h(comm)m(utes)1159 1092 y(Mor)1335 1107 y Fr(C)1380 1092 y Fx(\()p Ft(A;)17 b(C)7 b Fx(\))612 b Fu(F)10 b Fx(\()p Ft(C)d Fx(\))p 1678 1069 555 4 v 2150 1067 a Fl(-)1876 1136 y Fq( )r Fn(\()p Fq(C)e Fn(\))1158 604 y Fx(Mor)1334 619 y Fr(C)1379 604 y Fx(\()p Ft(A;)17 b(B)5 b Fx(\))610 b Fu(F)10 b Fx(\()p Ft(B)5 b Fx(\))p 1679 581 553 4 v 2149 579 a Fl(-)1875 543 y Fq( )r Fn(\()p Fq(B)s Fn(\))p 1403 999 4 351 v 1404 999 a Fl(?)1034 840 y Fn(Mor)1162 851 y Fh(C)1202 840 y Fn(\()p Fq(A;g)r Fn(\))p 2377 999 V 2379 999 a Fl(?)2418 840 y Fr(F)i Fn(\()p Fq(g)r Fn(\))0 1276 y Fx(since)33 b Ft( )t Fx(\()p Ft(C)7 b Fx(\)Mor)635 1291 y Fr(C)680 1276 y Fx(\()p Ft(A;)17 b(g)t Fx(\)\()p Ft(f)11 b Fx(\))26 b(=)i Ft( )t Fx(\()p Ft(C)7 b Fx(\)\()p Ft(g)t(f)k Fx(\))26 b(=)i Fu(F)10 b Fx(\()p Ft(g)t(f)h Fx(\)\()p Ft(a)p Fx(\))26 b(=)h Fu(F)10 b Fx(\()p Ft(g)t Fx(\))p Fu(F)g Fx(\()p Ft(f)h Fx(\)\()p Ft(a)p Fx(\))26 b(=)h Fu(F)10 b Fx(\()p Ft(g)t Fx(\))p Ft( )t Fx(\()p Ft(B)5 b Fx(\)\()p Ft(f)11 b Fx(\))p Ft(:)149 1393 y Fu(\()p Fx(:)31 b(Let)g Ft(A)h Fx(b)s(e)f(giv)m(en.)43 b(Let)31 b Ft(a)d Fx(:=)g Ft(')p Fx(\()p Ft(A)p Fx(\))1604 1356 y Fr(\000)p Fn(1)1698 1393 y Fx(\(1)1785 1408 y Fq(A)1842 1393 y Fx(\).)42 b(F)-8 b(or)31 b Ft(y)g Fu(2)d(F)10 b Fx(\()p Ft(B)5 b Fx(\))30 b(w)m(e)i(get)f Ft(y)g Fx(=)d Ft(')p Fx(\()p Ft(B)5 b Fx(\))3268 1356 y Fr(\000)p Fn(1)3362 1393 y Fx(\()p Ft(f)11 b Fx(\))27 b(=)0 1509 y Ft(')p Fx(\()p Ft(B)5 b Fx(\))219 1473 y Fr(\000)p Fn(1)313 1509 y Fx(\()p Ft(f)11 b Fx(1)459 1524 y Fq(A)516 1509 y Fx(\))29 b(=)g Ft(')p Fx(\()p Ft(B)5 b Fx(\))907 1473 y Fr(\000)p Fn(1)1001 1509 y Fx(Mor)1177 1524 y Fr(C)1222 1509 y Fx(\()p Ft(A;)17 b(f)11 b Fx(\)\(1)1561 1524 y Fq(A)1617 1509 y Fx(\))29 b(=)g Fu(F)10 b Fx(\()p Ft(f)h Fx(\))p Ft(')p Fx(\()p Ft(A)p Fx(\))2219 1473 y Fr(\000)p Fn(1)2313 1509 y Fx(\(1)2400 1524 y Fq(A)2456 1509 y Fx(\))29 b(=)g Fu(F)10 b Fx(\()p Ft(f)h Fx(\)\()p Ft(a)p Fx(\))33 b(for)g(a)g(uniquely)0 1625 y(determined)f Ft(f)39 b Fu(2)28 b Fx(Mor)866 1640 y Fr(C)911 1625 y Fx(\()p Ft(A;)17 b(B)5 b Fx(\).)p 3529 1625 4 66 v 3533 1562 59 4 v 3533 1625 V 3591 1625 4 66 v 149 1799 a Fv(Prop)s(osition)36 b(8.8.5.)49 b Fs(Given)37 b(a)g(r)-5 b(epr)g(esentable)36 b(functor)i Fu(F)2439 1814 y Fq(X)2538 1799 y Fx(:)32 b Fu(C)38 b(\000)-58 b(!)33 b Fv(Set)k Fs(for)g(e)-5 b(ach)37 b Ft(X)j Fu(2)0 1916 y(D)s Fs(.)46 b(Given)35 b(a)g(natur)-5 b(al)36 b(tr)-5 b(ansformation)34 b Fu(F)1599 1931 y Fq(g)1668 1916 y Fx(:)28 b Fu(F)1795 1931 y Fq(Y)1884 1916 y Fu(\000)-58 b(!)30 b(F)2105 1931 y Fq(X)2207 1916 y Fs(for)35 b(e)-5 b(ach)35 b Ft(g)d Fx(:)d Ft(X)36 b Fu(\000)-58 b(!)29 b Ft(Y)57 b Fs(\(c)-5 b(ontr)g(avari-)0 2032 y(ant!\))63 b(such)40 b(that)i Fu(F)50 b Fs(dep)-5 b(ends)40 b(functorial)5 b(ly)41 b(on)f Ft(X)8 b Fs(,)42 b(i.e.)63 b Fu(F)2299 2047 y Fn(1)2334 2058 y Fk(X)2434 2032 y Fx(=)39 b(1)2598 2047 y Fr(F)2648 2058 y Fk(X)2710 2032 y Ft(;)17 b Fu(F)2826 2047 y Fq(hg)2945 2032 y Fx(=)39 b Fu(F)3132 2047 y Fq(g)3172 2032 y Fu(F)3244 2047 y Fq(h)3288 2032 y Fs(.)63 b(Then)0 2148 y(the)47 b(r)-5 b(epr)g(esenting)46 b(obje)-5 b(cts)47 b Fx(\()p Ft(A)1174 2163 y Fq(X)1241 2148 y Ft(;)17 b(a)1336 2163 y Fq(X)1403 2148 y Fx(\))47 b Fs(for)g Fu(F)1728 2163 y Fq(X)1842 2148 y Fs(dep)-5 b(end)46 b(functorial)5 b(ly)47 b(on)f Ft(X)8 b Fs(,)50 b(i.e.)81 b(for)47 b(e)-5 b(ach)0 2264 y Ft(g)39 b Fx(:)d Ft(X)43 b Fu(\000)-58 b(!)36 b Ft(Y)60 b Fs(ther)-5 b(e)40 b(is)e(a)h(unique)g(homomorphism)e Ft(A)2069 2279 y Fq(g)2145 2264 y Fx(:)f Ft(A)2281 2279 y Fq(X)2384 2264 y Fu(\000)-58 b(!)36 b Ft(A)2612 2279 y Fq(Y)2712 2264 y Fx(\()p Fs(with)j Fu(F)3038 2279 y Fq(X)3105 2264 y Fx(\()p Ft(A)3216 2279 y Fq(g)3256 2264 y Fx(\)\()p Ft(a)3383 2279 y Fq(X)3451 2264 y Fx(\))c(=)0 2381 y Fu(F)72 2396 y Fq(g)112 2381 y Fx(\()p Ft(A)223 2396 y Fq(Y)284 2381 y Fx(\)\()p Ft(a)411 2396 y Fq(Y)471 2381 y Fx(\)\))g Fs(and)f(the)h(fol)5 b(lowing)34 b(identities)g(hold)g Ft(A)2052 2396 y Fn(1)2087 2407 y Fk(X)2177 2381 y Fx(=)28 b(1)2330 2396 y Fq(A)2383 2407 y Fk(X)2445 2381 y Ft(;)17 b(A)2562 2396 y Fq(hg)2670 2381 y Fx(=)28 b Ft(A)2847 2396 y Fq(h)2891 2381 y Ft(A)2964 2396 y Fq(g)3005 2381 y Fs(.)149 2555 y Fi(Pr)n(oof.)42 b Fx(Cho)s(ose)e(a)e(represen)m(ting) i(ob)5 b(ject)40 b(\()p Ft(A)1937 2570 y Fq(X)2004 2555 y Ft(;)17 b(a)2099 2570 y Fq(X)2166 2555 y Fx(\))39 b(for)f Fu(F)2470 2570 y Fq(X)2576 2555 y Fx(for)g(eac)m(h)i Ft(X)45 b Fu(2)39 b(C)45 b Fx(\(b)m(y)40 b(the)0 2671 y(axiom)31 b(of)h(c)m(hoice\).)44 b(Then)34 b(there)f(is)f(a)g(unique)h (homomorphism)d Ft(A)2529 2686 y Fq(g)2597 2671 y Fx(:)e Ft(A)2725 2686 y Fq(X)2820 2671 y Fu(\000)-60 b(!)28 b Ft(A)3038 2686 y Fq(Y)3131 2671 y Fx(with)970 2828 y Fu(F)1042 2843 y Fq(X)1109 2828 y Fx(\()p Ft(A)1220 2843 y Fq(g)1260 2828 y Fx(\)\()p Ft(a)1387 2843 y Fq(X)1454 2828 y Fx(\))g(=)f Fu(F)1695 2843 y Fq(g)1735 2828 y Fx(\()p Ft(A)1846 2843 y Fq(Y)1907 2828 y Fx(\)\()p Ft(a)2034 2843 y Fq(Y)2095 2828 y Fx(\))g Fu(2)h(F)2326 2843 y Fq(X)2393 2828 y Fx(\()p Ft(A)2504 2843 y Fq(Y)2565 2828 y Fx(\))p Ft(;)0 2985 y Fx(for)45 b(eac)m(h)i Ft(g)54 b Fx(:)c Ft(X)59 b Fu(\000)-60 b(!)50 b Ft(Y)67 b Fx(b)s(ecause)47 b Fu(F)1450 3000 y Fq(g)1490 2985 y Fx(\()p Ft(A)1601 3000 y Fq(Y)1662 2985 y Fx(\))j(:)h Fu(F)1900 3000 y Fq(Y)1960 2985 y Fx(\()p Ft(A)2071 3000 y Fq(Y)2132 2985 y Fx(\))f Fu(\000)-60 b(!)51 b(F)2460 3000 y Fq(X)2527 2985 y Fx(\()p Ft(A)2638 3000 y Fq(Y)2698 2985 y Fx(\))46 b(is)g(giv)m(en.)83 b(W)-8 b(e)47 b(ha)m(v)m(e)0 3101 y Fu(F)72 3116 y Fq(X)139 3101 y Fx(\()p Ft(A)250 3116 y Fn(1)289 3101 y Fx(\)\()p Ft(a)416 3116 y Fq(X)484 3101 y Fx(\))34 b(=)g Fu(F)738 3116 y Fn(1)777 3101 y Fx(\()p Ft(A)888 3116 y Fq(X)955 3101 y Fx(\)\()p Ft(a)1082 3116 y Fq(X)1150 3101 y Fx(\))g(=)g Ft(a)1383 3116 y Fq(X)1485 3101 y Fx(=)g Fu(F)1667 3116 y Fq(X)1733 3101 y Fx(\(1\)\()p Ft(a)1947 3116 y Fq(X)2015 3101 y Fx(\))i(hence)i Ft(A)2437 3116 y Fn(1)2510 3101 y Fx(=)c(1,)j(and)g Fu(F)2999 3116 y Fq(X)3066 3101 y Fx(\()p Ft(A)3177 3116 y Fq(hg)3258 3101 y Fx(\)\()p Ft(a)3385 3116 y Fq(X)3452 3101 y Fx(\))d(=)0 3217 y Fu(F)72 3232 y Fq(hg)152 3217 y Fx(\()p Ft(A)263 3232 y Fq(Z)320 3217 y Fx(\)\()p Ft(a)447 3232 y Fq(Z)504 3217 y Fx(\))j(=)f Fu(F)763 3232 y Fq(g)803 3217 y Fx(\()p Ft(A)914 3232 y Fq(Z)971 3217 y Fx(\))p Fu(F)1081 3232 y Fq(h)1125 3217 y Fx(\()p Ft(A)1236 3232 y Fq(Z)1293 3217 y Fx(\)\()p Ft(a)1420 3232 y Fq(Z)1477 3217 y Fx(\))g(=)h Fu(F)1736 3232 y Fq(g)1776 3217 y Fx(\()p Ft(A)1887 3232 y Fq(Z)1944 3217 y Fx(\))p Fu(F)2054 3232 y Fq(Y)2114 3217 y Fx(\()p Ft(A)2225 3232 y Fq(h)2270 3217 y Fx(\)\()p Ft(a)2397 3232 y Fq(Y)2458 3217 y Fx(\))f(=)h Fu(F)2717 3232 y Fq(X)2784 3217 y Fx(\()p Ft(A)2895 3232 y Fq(h)2940 3217 y Fx(\))p Fu(F)3050 3232 y Fq(g)3090 3217 y Fx(\()p Ft(A)3201 3232 y Fq(Y)3262 3217 y Fx(\)\()p Ft(a)3389 3232 y Fq(Y)3449 3217 y Fx(\))g(=)0 3334 y Fu(F)72 3349 y Fq(X)139 3334 y Fx(\()p Ft(A)250 3349 y Fq(h)295 3334 y Fx(\))p Fu(F)405 3349 y Fq(X)472 3334 y Fx(\()p Ft(A)583 3349 y Fq(g)623 3334 y Fx(\)\()p Ft(a)750 3349 y Fq(X)817 3334 y Fx(\))k(=)g Fu(F)1085 3349 y Fq(X)1152 3334 y Fx(\()p Ft(A)1263 3349 y Fq(h)1308 3334 y Ft(A)1381 3349 y Fq(g)1421 3334 y Fx(\)\()p Ft(a)1548 3349 y Fq(X)1616 3334 y Fx(\))f(hence)i Ft(A)2046 3349 y Fq(h)2091 3334 y Ft(A)2164 3349 y Fq(g)2245 3334 y Fx(=)f Ft(A)2435 3349 y Fq(hg)2556 3334 y Fx(for)f Ft(g)k Fx(:)d Ft(X)49 b Fu(\000)-60 b(!)41 b Ft(Y)61 b Fx(and)41 b Ft(h)g Fx(:)0 3450 y Ft(Y)49 b Fu(\000)-60 b(!)28 b Ft(Z)39 b Fx(in)32 b Fu(D)s Fx(.)p 3529 3450 V 3533 3387 59 4 v 3533 3450 V 3591 3450 4 66 v 149 3624 a Fv(Corollary)k(8.8.6.)50 b Fx(1.)60 b(Map\()p Ft(X)r(;)17 b(M)10 b Fx(\))1631 3597 y Fu(\030)1632 3628 y Fx(=)1746 3624 y(Hom)1949 3639 y Fq(R)2007 3624 y Fx(\()p Ft(R)q(X)r(;)17 b(M)10 b Fx(\))41 b Fs(is)f(a)g(natur)-5 b(al)40 b(tr)-5 b(ansformation)0 3740 y(in)35 b Ft(M)45 b Fs(\(and)34 b(in)h Ft(X)8 b Fs(!\).)44 b(In)34 b(p)-5 b(articular)35 b Fv(Set)28 b Fu(3)g Ft(X)35 b Fu(7!)28 b Ft(R)q(X)35 b Fu(2)28 b Ft(R)q Fs(-)p Fv(Mo)s(d)36 b Fs(is)e(a)h(functor.)149 3857 y Fx(2.)57 b(Bil)405 3872 y Fq(R)463 3857 y Fx(\()p Ft(M)5 b(;)17 b(N)10 b Fx(;)17 b Ft(A)p Fx(\))922 3829 y Fu(\030)923 3861 y Fx(=)1034 3857 y(Hom\()p Ft(M)35 b Fu(\012)1482 3872 y Fq(R)1565 3857 y Ft(N)5 b(;)17 b(A)p Fx(\))39 b Fs(is)g(a)f(natur)-5 b(al)39 b(tr)-5 b(ansformation)38 b(in)g Ft(A)h Fs(\(and)f(in)0 3973 y Fx(\()p Ft(M)5 b(;)17 b(N)10 b Fx(\))28 b Fu(2)g Fv(Mo)s(d)q Fs(-)p Ft(R)14 b Fu(\002)f Ft(R)q Fs(-)p Fv(Mo)s(d)q Fx(\))p Fs(.)43 b(In)30 b(p)-5 b(articular)31 b Fv(Mo)s(d)q Fs(-)p Ft(R)14 b Fu(\002)f Ft(R)q Fs(-)p Fv(Mo)s(d)29 b Fu(3)f Ft(M)5 b(;)17 b(N)38 b Fu(7!)27 b Ft(M)d Fu(\012)3366 3988 y Fq(r)3417 3973 y Ft(N)39 b Fu(2)0 4089 y Fv(Ab)c Fs(is)f(a)h(functor.)149 4205 y Fx(3.)45 b Ft(R)q Fs(-)p Fv(Mo)s(d)q Fs(-)p Ft(S)28 b Fu(\002)22 b Ft(S)6 b Fs(-)p Fv(Mo)s(d)q Fs(-)p Ft(T)41 b Fu(3)28 b Fx(\()p Ft(M)5 b(;)17 b(N)10 b Fx(\))28 b Fu(7!)f Ft(M)33 b Fu(\012)2052 4220 y Fq(S)2126 4205 y Ft(N)38 b Fu(2)28 b Ft(R)q Fs(-)p Fv(Mo)s(d)q Fs(-)p Ft(T)48 b Fs(is)35 b(a)f(functor.)p eop %%Page: 40 40 40 39 bop 0 236 a Fo(40)1456 b(8.)77 b(TOOLBO)n(X)690 425 y Fv(9.)93 b(Adjoin)m(t)37 b(F)-9 b(unctors)37 b(and)h(the)g(Y)-9 b(oneda)38 b(Lemma)149 600 y(Theorem)g(8.9.1.)49 b Fs(\(Y)-7 b(one)i(da)24 b(L)-5 b(emma\))24 b(L)-5 b(et)26 b Fu(C)31 b Fs(b)-5 b(e)25 b(a)g(c)-5 b(ate)g(gory.)41 b(Given)25 b(a)g(c)-5 b(ovariant)25 b(functor)0 716 y Fu(F)37 b Fx(:)28 b Fu(C)34 b(\000)-58 b(!)28 b Fv(Set)35 b Fs(and)f(an)h(obje)-5 b(ct)34 b Ft(A)28 b Fu(2)g(C)6 b Fs(.)45 b(Then)34 b(the)h(map)787 908 y Ft(\031)d Fx(:)27 b(Nat\(Mor)1302 923 y Fr(C)1347 908 y Fx(\()p Ft(A;)17 b Fs(-)p Fx(\))p Ft(;)g Fu(F)10 b Fx(\))27 b Fu(3)h Ft(\036)f Fu(7!)g Ft(\036)p Fx(\()p Ft(A)p Fx(\)\(1)2366 923 y Fq(A)2423 908 y Fx(\))g Fu(2)i(F)10 b Fx(\()p Ft(A)p Fx(\))0 1100 y Fs(is)35 b(bije)-5 b(ctive)34 b(with)g(the)h(inverse)f(map)875 1292 y Ft(\031)934 1250 y Fr(\000)p Fn(1)1056 1292 y Fx(:)27 b Fu(F)10 b Fx(\()p Ft(A)p Fx(\))28 b Fu(3)g Ft(a)g Fu(7!)f Ft(h)1725 1250 y Fq(a)1794 1292 y Fu(2)h Fx(Nat\(Mor)2262 1307 y Fr(C)2307 1292 y Fx(\()p Ft(A;)17 b Fs(-)p Fx(\))p Ft(;)g Fu(F)10 b Fx(\))p Ft(;)0 1484 y Fs(wher)-5 b(e)34 b Ft(h)331 1447 y Fq(a)373 1484 y Fx(\()p Ft(B)5 b Fx(\)\()p Ft(f)11 b Fx(\))27 b(=)h Fu(F)10 b Fx(\()p Ft(f)h Fx(\)\()p Ft(a)p Fx(\))p Fs(.)149 1678 y Fi(Pr)n(oof.)42 b Fx(F)-8 b(or)32 b Ft(\036)27 b Fu(2)h Fx(Nat\(Mor)1262 1693 y Fr(C)1307 1678 y Fx(\()p Ft(A;)17 b Fx(-\))p Ft(;)g Fu(F)10 b Fx(\))31 b(w)m(e)i(ha)m(v)m(e)h(a)e(map)g Ft(\036)p Fx(\()p Ft(A)p Fx(\))27 b(:)h(Mor)2859 1693 y Fr(C)2904 1678 y Fx(\()p Ft(A;)17 b(A)p Fx(\))28 b Fu(\000)-60 b(!)27 b(F)10 b Fx(\()p Ft(A)p Fx(\),)0 1794 y(hence)33 b Ft(\031)i Fx(with)d Ft(\031)t Fx(\()p Ft(\036)p Fx(\))27 b(:=)g Ft(\036)p Fx(\()p Ft(A)p Fx(\)\(1)1226 1809 y Fq(A)1283 1794 y Fx(\))k(is)g(a)h(w)m(ell)f(de\014ned)i(map.)42 b(F)-8 b(or)31 b Ft(\031)2549 1758 y Fr(\000)p Fn(1)2675 1794 y Fx(w)m(e)h(ha)m(v)m(e)h(to)e(c)m(hec)m(k)j(that)0 1910 y Ft(h)56 1874 y Fq(a)130 1910 y Fx(is)e(a)h(natural)e(transformation.) 41 b(Giv)m(en)33 b Ft(f)38 b Fx(:)28 b Ft(B)33 b Fu(\000)-60 b(!)28 b Ft(C)39 b Fx(in)32 b Fu(C)6 b Fx(.)44 b(Then)33 b(the)g(diagram)1195 2612 y Fu(F)10 b Fx(\()p Ft(B)5 b Fx(\))739 b Fu(F)10 b Fx(\()p Ft(C)d Fx(\))p 1460 2589 682 4 v 2059 2587 a Fl(-)1724 2656 y Fr(F)g Fn(\()p Fq(f)g Fn(\))1067 2124 y Fx(Mor)1243 2139 y Fr(C)1288 2124 y Fx(\()p Ft(A;)17 b(B)5 b Fx(\))483 b(Mor)2219 2139 y Fr(C)2264 2124 y Fx(\()p Ft(A;)17 b(C)7 b Fx(\))p 1588 2101 427 4 v 1932 2099 a Fl(-)1652 2064 y Fn(Mor\()p Fq(A;f)g Fn(\))p 1311 2519 4 351 v 1313 2519 a Fl(?)1084 2360 y Fq(h)1125 2336 y Fk(a)1163 2360 y Fn(\()p Fq(B)s Fn(\))p 2286 2519 V 2288 2519 a Fl(?)2327 2360 y Fq(h)2368 2336 y Fk(a)2405 2360 y Fn(\()p Fq(C)e Fn(\))0 2832 y Fx(is)34 b(comm)m(utativ)m(e.)49 b(In)35 b(fact)g(if)f Ft(g)g Fu(2)e Fx(Mor)1493 2847 y Fr(C)1538 2832 y Fx(\()p Ft(A;)17 b(B)5 b Fx(\))35 b(then)g Ft(h)2125 2795 y Fq(a)2167 2832 y Fx(\()p Ft(C)7 b Fx(\)Mor)2496 2847 y Fr(C)2541 2832 y Fx(\()p Ft(A;)17 b(f)11 b Fx(\)\()p Ft(g)t Fx(\))30 b(=)h Ft(h)3113 2795 y Fq(a)3155 2832 y Fx(\()p Ft(C)7 b Fx(\)\()p Ft(f)k(g)t Fx(\))30 b(=)0 2948 y Fu(F)10 b Fx(\()p Ft(f)h(g)t Fx(\)\()p Ft(a)p Fx(\))26 b(=)i Fu(F)10 b Fx(\()p Ft(f)h Fx(\))p Fu(F)f Fx(\()p Ft(g)t Fx(\)\()p Ft(a)p Fx(\))25 b(=)j Fu(F)10 b Fx(\()p Ft(f)h Fx(\))p Ft(h)1480 2912 y Fq(a)1521 2948 y Fx(\()p Ft(B)5 b Fx(\)\()p Ft(a)p Fx(\).)43 b(Th)m(us)34 b Ft(\031)2179 2912 y Fr(\000)p Fn(1)2306 2948 y Fx(is)e(w)m(ell)g(de\014ned.)149 3064 y(Let)45 b Ft(\031)395 3028 y Fr(\000)p Fn(1)490 3064 y Fx(\()p Ft(a)p Fx(\))j(=)h Ft(h)846 3028 y Fq(a)887 3064 y Fx(.)80 b(Then)46 b Ft(\031)t(\031)1379 3028 y Fr(\000)p Fn(1)1474 3064 y Fx(\()p Ft(a)p Fx(\))i(=)g Ft(h)1829 3028 y Fq(a)1871 3064 y Fx(\()p Ft(A)p Fx(\)\(1)2107 3079 y Fq(A)2164 3064 y Fx(\))g(=)h Fu(F)10 b Fx(\(1)2544 3079 y Fq(A)2600 3064 y Fx(\)\()p Ft(a)p Fx(\))48 b(=)h Ft(a)p Fx(.)80 b(Let)45 b Ft(\031)t Fx(\()p Ft(\036)p Fx(\))j(=)0 3180 y Ft(\036)p Fx(\()p Ft(A)p Fx(\)\(1)294 3195 y Fq(A)350 3180 y Fx(\))38 b(=)g Ft(a)p Fx(.)62 b(Then)40 b Ft(\031)1000 3144 y Fr(\000)p Fn(1)1094 3180 y Ft(\031)t Fx(\()p Ft(\036)p Fx(\))e(=)g Ft(h)1495 3144 y Fq(a)1575 3180 y Fx(and)h Ft(h)1827 3144 y Fq(a)1869 3180 y Fx(\()p Ft(B)5 b Fx(\)\()p Ft(f)11 b Fx(\))38 b(=)g Fu(F)10 b Fx(\()p Ft(f)h Fx(\)\()p Ft(a)p Fx(\))37 b(=)h Fu(F)10 b Fx(\()p Ft(f)h Fx(\)\()p Ft(\036)p Fx(\()p Ft(A)p Fx(\)\(1)3355 3195 y Fq(A)3410 3180 y Fx(\)\))38 b(=)0 3296 y Ft(\036)p Fx(\()p Ft(B)5 b Fx(\)Mor)389 3311 y Fr(C)434 3296 y Fx(\()p Ft(A;)17 b(f)11 b Fx(\)\(1)773 3311 y Fq(A)829 3296 y Fx(\))28 b(=)f Ft(\036)p Fx(\()p Ft(B)5 b Fx(\)\()p Ft(f)11 b Fx(\),)32 b(also)g Ft(h)1657 3260 y Fq(a)1726 3296 y Fx(=)c Ft(\036)p Fx(.)p 3529 3296 4 66 v 3533 3234 59 4 v 3533 3296 V 3591 3296 4 66 v 149 3491 a Fv(Corollary)36 b(8.9.2.)50 b Fs(Given)34 b Ft(A;)17 b(B)33 b Fu(2)28 b(C)6 b Fs(.)45 b(Then)34 b(the)h(fol)5 b(lowing)33 b(hold)149 3607 y Fx(1.)81 b(Mor)482 3622 y Fr(C)527 3607 y Fx(\()p Ft(A;)17 b(B)5 b Fx(\))49 b Fu(3)h Ft(f)61 b Fu(7!)49 b Fx(Mor)1398 3622 y Fr(C)1443 3607 y Fx(\()p Ft(f)5 b(;)17 b Fs(-)p Fx(\))49 b Fu(2)h Fx(Nat\(Mor)2190 3622 y Fr(C)2235 3607 y Fx(\()p Ft(B)5 b(;)17 b Fs(-)p Fx(\))p Ft(;)g Fx(Mor)2689 3622 y Fr(C)2734 3607 y Fx(\()p Ft(A;)g Fs(-)o Fx(\)\))47 b Fs(is)f(a)h(bije)-5 b(ctive)0 3723 y(map.)149 3840 y Fx(2.)45 b Fs(With)35 b(the)f(bije)-5 b(ctive)33 b(map)h(fr)-5 b(om)33 b(1.)45 b(the)34 b(isomorphisms)e(fr)-5 b(om)34 b Fx(Mor)2798 3855 y Fr(C)2843 3840 y Fx(\()p Ft(A;)17 b(B)5 b Fx(\))34 b Fs(c)-5 b(orr)g(esp)g(ond)0 3956 y(to)35 b(natur)-5 b(al)35 b(isomorphisms)e(fr)-5 b(om)34 b Fx(Nat\(Mor)1670 3971 y Fr(C)1715 3956 y Fx(\()p Ft(B)5 b(;)17 b Fs(-)p Fx(\))p Ft(;)g Fx(Mor)2169 3971 y Fr(C)2214 3956 y Fx(\()p Ft(A;)g Fs(-)o Fx(\)\))p Fs(.)149 4072 y Fx(3.)45 b Fs(F)-7 b(or)34 b(c)-5 b(ontr)g(avariant)34 b(functors)h Fu(F)i Fx(:)28 b Fu(C)34 b(\000)-58 b(!)28 b Fv(Set)35 b Fs(we)g(have)f Fx(Nat\(Mor)2764 4087 y Fr(C)2809 4072 y Fx(\()p Fs(-)o Ft(;)17 b(A)p Fx(\))p Ft(;)g Fu(F)10 b Fx(\))3227 4044 y Fu(\030)3228 4076 y Fx(=)3332 4072 y Fu(F)g Fx(\()p Ft(A)p Fx(\))p Fs(.)149 4188 y Fx(4.)44 b(Mor)445 4203 y Fr(C)490 4188 y Fx(\()p Ft(A;)17 b(B)5 b Fx(\))27 b Fu(3)i Ft(f)38 b Fu(7!)27 b Fx(Mor)1273 4203 y Fr(C)1318 4188 y Fx(\()p Fs(-)p Ft(;)17 b(f)11 b Fx(\))27 b Fu(2)h Fx(Nat\(Mor)2027 4203 y Fr(C)2072 4188 y Fx(\()p Fs(-)p Ft(;)17 b(A)p Fx(\))p Ft(;)g Fx(Mor)2520 4203 y Fr(C)2565 4188 y Fx(\()p Fs(-)p Ft(;)g(B)5 b Fx(\)\))30 b Fs(is)h(a)g(bije)-5 b(ctive)30 b(map)0 4304 y(that)j(de\014nes)f(a)g(one-to-one)g(c)-5 b(orr)g(esp)g(ondenc)g(e)31 b(b)-5 b(etwe)g(en)32 b(the)h(isomorphisms) e(fr)-5 b(om)32 b Fx(Mor)3283 4319 y Fr(C)3328 4304 y Fx(\()p Ft(A;)17 b(B)5 b Fx(\))0 4421 y Fs(and)34 b(the)h(natur)-5 b(al)35 b(isomorphisms)e(fr)-5 b(om)35 b Fx(Nat)o(\(Mor)1904 4436 y Fr(C)1949 4421 y Fx(\()p Fs(-)p Ft(;)17 b(A)p Fx(\))p Ft(;)g Fx(Mor)2397 4436 y Fr(C)2442 4421 y Fx(\()p Fs(-)p Ft(;)g(B)5 b Fx(\)\))p Fs(.)149 4615 y Fi(Pr)n(oof.)42 b Fx(1.)i(follo)m(ws)31 b(from)g(the)i(Y)-8 b(oneda)33 b(Lemma)e(with)h Fu(F)38 b Fx(=)27 b(Mor)2678 4630 y Fr(C)2723 4615 y Fx(\()p Ft(A;)17 b Fx(-\).)149 4731 y(2.)69 b(Observ)m(e)43 b(that)d Ft(h)952 4695 y Fq(f)998 4731 y Fx(\()p Ft(C)7 b Fx(\)\()p Ft(g)t Fx(\))41 b(=)g(Mor)1612 4746 y Fr(C)1658 4731 y Fx(\()p Ft(A;)17 b(g)t Fx(\)\()p Ft(f)11 b Fx(\))40 b(=)i Ft(g)t(f)52 b Fx(=)41 b(Mor)2639 4746 y Fr(C)2685 4731 y Fx(\()p Ft(f)5 b(;)17 b(C)7 b Fx(\)\()p Ft(g)t Fx(\))39 b(hence)k Ft(h)3437 4695 y Fq(f)3524 4731 y Fx(=)0 4848 y(Mor)176 4863 y Fr(C)221 4848 y Fx(\()p Ft(f)5 b(;)17 b Fx(-\).)39 b(Since)22 b(w)m(e)h(ha)m(v)m(e)g(Mor)1260 4863 y Fr(C)1305 4848 y Fx(\()p Ft(f)5 b(;)17 b Fx(-\)Mor)1687 4863 y Fr(C)1732 4848 y Fx(\()p Ft(g)t(;)g Fx(-)n(\))28 b(=)f(Mor)2242 4863 y Fr(C)2287 4848 y Fx(\()p Ft(g)t(f)5 b(;)17 b Fx(-)n(\))22 b(and)g(Mor)2919 4863 y Fr(C)2964 4848 y Fx(\()p Ft(f)5 b(;)17 b Fx(-)o(\))28 b(=)g(id)3382 4863 y Fn(Mor)3509 4874 y Fh(C)3549 4863 y Fn(\()p Fq(A;)p Fx(-)p Fn(\))0 4964 y Fx(if)e(and)i(only)e(if)g Ft(f)39 b Fx(=)27 b(1)800 4979 y Fq(A)884 4964 y Fx(w)m(e)i(get)e(the)h(one-to-one)e(corresp)s (ondence)k(b)s(et)m(w)m(een)f(the)f(isomorphisms)0 5080 y(from)j(1.)149 5196 y(3.)44 b(and)32 b(4.)44 b(follo)m(w)30 b(b)m(y)k(dualizing.)p 3529 5196 V 3533 5134 59 4 v 3533 5196 V 3591 5196 4 66 v eop %%Page: 41 41 41 40 bop 777 236 a Fo(9.)76 b(ADJOINT)33 b(FUNCTORS)f(AND)g(THE)i (YONED)n(A)d(LEMMA)675 b(41)149 425 y Fv(Remark)37 b(8.9.3.)50 b Fx(The)33 b(map)f Ft(\031)37 b Fx(is)32 b(a)h(natural)e (transformation)g(in)h(the)h(argumen)m(ts)g Ft(A)g Fx(and)0 541 y Fu(F)10 b Fx(.)76 b(More)44 b(precisely:)66 b(if)42 b Ft(f)57 b Fx(:)47 b Ft(A)f Fu(\000)-60 b(!)46 b Ft(B)j Fx(and)44 b Ft(\036)i Fx(:)g Fu(F)56 b(\000)-60 b(!)47 b(G)i Fx(are)44 b(giv)m(en)f(then)i(the)f(follo)m(wing)0 658 y(diagrams)31 b(comm)m(ute)1199 1252 y(Nat\(Mor)1573 1267 y Fr(C)1618 1252 y Fx(\()p Ft(A;)17 b Fx(-\))p Ft(;)g Fu(G)6 b Fx(\))277 b Fu(G)6 b Fx(\()p Ft(A)p Fx(\))p 2019 1229 221 4 v 2157 1227 a Fl(-)2108 1275 y Fq(\031)1191 765 y Fx(Nat\(Mor)1565 780 y Fr(C)1610 765 y Fx(\()p Ft(A;)17 b Fx(-)o(\))p Ft(;)g Fu(F)10 b Fx(\))260 b Fu(F)10 b Fx(\()p Ft(A)p Fx(\))p 2028 741 203 4 v 2148 739 a Fl(-)2108 721 y Fq(\031)p 1593 1159 4 351 v 1595 1159 a Fl(?)1036 1000 y Fn(Nat)q(\(Mor\()p Fq(A;)p Fx(-)o Fn(\))p Fq(;\036)p Fn(\))p 2373 1159 V 2375 1159 a Fl(?)2414 1000 y Fq(\036)p Fn(\()p Fq(A)p Fn(\))1186 1877 y Fx(Nat\(Mor)1560 1892 y Fr(C)1605 1877 y Fx(\()p Ft(B)5 b(;)17 b Fx(-)o(\))p Ft(;)g Fu(F)10 b Fx(\))241 b Fu(F)10 b Fx(\()p Ft(B)5 b Fx(\))p Ft(:)p 2029 1854 184 4 v 2130 1852 a Fl(-)2099 1900 y Fq(\031)1189 1390 y Fx(Nat\(Mor)1563 1405 y Fr(C)1608 1390 y Fx(\()p Ft(A;)17 b Fx(-)o(\))p Ft(;)g Fu(F)10 b Fx(\))261 b Fu(F)10 b Fx(\()p Ft(A)p Fx(\))p 2026 1366 203 4 v 2146 1364 a Fl(-)2106 1346 y Fq(\031)p 1591 1784 4 351 v 1593 1784 a Fl(?)1035 1625 y Fn(Nat)q(\(Mor)o(\()p Fq(f)s(;)p Fx(-)p Fn(\))p Fq(;)p Fr(F)d Fn(\))p 2371 1784 V 2373 1784 a Fl(?)2412 1625 y Fr(F)g Fn(\()p Fq(f)g Fn(\))0 2010 y Fx(This)33 b(can)g(b)s(e)f(easily)g(c)m(hec)m(k)m(ed.)47 b(F)-8 b(urthermore)32 b(w)m(e)h(ha)m(v)m(e)h(for)e Ft( )g Fx(:)c(Mor)2607 2025 y Fr(C)2652 2010 y Fx(\()p Ft(A;)17 b Fx(-)o(\))28 b Fu(\000)-60 b(!)28 b(F)135 2171 y Ft(\031)20 b Fx(Nat\(Mor)584 2186 y Fr(C)629 2171 y Fx(\()p Ft(A;)d Fx(-\))p Ft(;)g(\036)p Fx(\)\()p Ft( )t Fx(\))26 b(=)i Ft(\031)t Fx(\()p Ft(\036 )t Fx(\))f(=)h(\()p Ft(\036 )t Fx(\)\()p Ft(A)p Fx(\)\(1)2096 2186 y Fq(A)2152 2171 y Fx(\))g(=)f Ft(\036)p Fx(\()p Ft(A)p Fx(\))p Ft( )t Fx(\()p Ft(A)p Fx(\)\(1)2831 2186 y Fq(A)2888 2171 y Fx(\))g(=)h Ft(\036)p Fx(\()p Ft(A)p Fx(\))p Ft(\031)t Fx(\()p Ft( )t Fx(\))0 2333 y(and)153 2468 y Ft(\031)20 b Fx(Nat\(Mor)602 2483 y Fr(C)647 2468 y Fx(\()p Ft(f)5 b(;)17 b Fx(-\))p Ft(;)g Fu(F)10 b Fx(\)\()p Ft( )t Fx(\))26 b(=)i Ft(\031)t Fx(\()p Ft( )t Fx(Mor)1630 2483 y Fr(C)1675 2468 y Fx(\()p Ft(f)5 b(;)17 b Fx(-)o(\)\))28 b(=)g(\()p Ft( )t Fx(Mor)2331 2483 y Fr(C)2376 2468 y Fx(\()p Ft(f)5 b(;)17 b Fx(-)o(\)\)\()p Ft(B)5 b Fx(\)\(1)2861 2483 y Fq(B)2921 2468 y Fx(\))28 b(=)f Ft( )t Fx(\()p Ft(B)5 b Fx(\)\()p Ft(f)11 b Fx(\))153 2584 y(=)27 b Ft( )t Fx(\()p Ft(B)5 b Fx(\)Mor)655 2599 y Fr(C)700 2584 y Fx(\()p Ft(A;)17 b(f)11 b Fx(\)\(1)1039 2599 y Fq(A)1095 2584 y Fx(\))27 b(=)h Fu(F)10 b Fx(\()p Ft(f)h Fx(\))p Ft( )t Fx(\()p Ft(A)p Fx(\)\(1)1784 2599 y Fq(A)1840 2584 y Fx(\))27 b(=)h Fu(F)10 b Fx(\()p Ft(f)h Fx(\))p Ft(\031)t Fx(\()p Ft( )t Fx(\))p Ft(:)149 2757 y Fv(Remark)37 b(8.9.4.)50 b Fx(By)35 b(the)h(previous)f(corollary)e(the)j(represen)m (ting)g(ob)5 b(ject)35 b Ft(A)g Fx(is)g(uniquely)0 2874 y(determined)d(up)h(to)g(isomorphism)d(b)m(y)j(the)g(isomorphism)d (class)j(of)f(the)h(functor)g(Mor)3236 2889 y Fr(C)3281 2874 y Fx(\()p Ft(A;)17 b Fx(-)o(\).)149 3051 y Fv(Problem)36 b(8.9.15.)49 b Fx(1.)42 b(Determine)28 b(explicitly)f(all)g(natural)h (endomorphisms)g(from)f Fp(G)3437 3066 y Fq(a)3513 3051 y Fx(to)0 3167 y Fp(G)71 3182 y Fq(a)152 3167 y Fx(\(as)32 b(de\014ned)i(in)e(Lemma)f(2.3.5\).)149 3283 y(2.)44 b(Determine)31 b(all)g(additiv)m(e)h(natural)f(endomorphisms)h(of)g Fp(G)2472 3298 y Fq(a)2520 3283 y Fx(.)149 3399 y(3.)44 b(Determine)31 b(all)g(natural)g(transformations)g(from)g Fp(G)2226 3414 y Fq(a)2306 3399 y Fx(to)h Fp(G)2497 3414 y Fq(m)2602 3399 y Fx(\(see)h(Lemma)f(2.3.7\).)149 3516 y(4.)44 b(Determine)31 b(all)g(natural)g(automorphisms)g(of)h Fp(G)2088 3531 y Fq(m)2160 3516 y Fx(.)149 3693 y Fv(Prop)s(osition)k (8.9.5.)49 b Fs(L)-5 b(et)46 b Fu(G)52 b Fx(:)47 b Fu(C)35 b(\002)c(D)49 b(\000)-58 b(!)46 b Fv(Set)f Fs(b)-5 b(e)45 b(a)g(c)-5 b(ovariant)44 b(bifunctor)h(such)g(that)0 3809 y(the)c(functor)g Fu(G)6 b Fx(\()p Ft(C)r(;)17 b Fs(-)p Fx(\))39 b(:)g Fu(D)j(\000)-58 b(!)40 b Fv(Set)h Fs(is)g(r)-5 b(epr)g(esentable)40 b(for)h(al)5 b(l)40 b Ft(C)47 b Fu(2)39 b(C)6 b Fs(.)64 b(Then)40 b(ther)-5 b(e)41 b(exists)g(a)0 3925 y(c)-5 b(ontr)g(avariant)37 b(functor)g Fu(F)42 b Fx(:)33 b Fu(C)38 b(\000)-58 b(!)33 b(D)40 b Fs(such)d(that)h Fu(G)2002 3898 y(\030)2002 3929 y Fx(=)2111 3925 y(Mor)2288 3940 y Fr(D)2348 3925 y Fx(\()p Fu(F)10 b Fs(-)p Ft(;)17 b Fs(-)o Fx(\))37 b Fs(holds.)52 b(F)-7 b(urthermor)i(e)37 b Fu(F)0 4041 y Fs(is)e(uniquely)g(determine)-5 b(d)34 b(by)h Fu(G)41 b Fs(up)35 b(to)g(isomorphism.)149 4218 y Fi(Pr)n(oof.)42 b Fx(F)-8 b(or)39 b(eac)m(h)h Ft(C)46 b Fu(2)39 b(C)45 b Fx(c)m(ho)s(ose)40 b(an)f(ob)5 b(ject)40 b Fu(F)10 b Fx(\()p Ft(C)d Fx(\))38 b Fu(2)i(D)h Fx(and)f(an)f(isomorphism)d Ft(\030)3475 4233 y Fq(C)3573 4218 y Fx(:)0 4335 y Fu(G)6 b Fx(\()p Ft(C)r(;)17 b Fx(-)o(\))326 4307 y Fu(\030)327 4339 y Fx(=)441 4335 y(Mor)617 4350 y Fr(D)678 4335 y Fx(\()p Fu(F)10 b Fx(\()p Ft(C)d Fx(\))p Ft(;)17 b Fx(-)n(\).)61 b(Giv)m(en)38 b Ft(f)48 b Fx(:)38 b Ft(C)44 b Fu(\000)-60 b(!)37 b Ft(C)1946 4298 y Fr(0)2008 4335 y Fx(in)g Fu(C)45 b Fx(then)39 b(let)e Fu(F)10 b Fx(\()p Ft(f)h Fx(\))37 b(:)g Fu(F)10 b Fx(\()p Ft(C)3113 4298 y Fr(0)3136 4335 y Fx(\))37 b Fu(\000)-60 b(!)38 b(F)10 b Fx(\()p Ft(C)d Fx(\))0 4451 y(b)s(e)38 b(the)h(uniquely)f(determined)g(morphism)e(\(b) m(y)j(the)f(Y)-8 b(oneda)39 b(Lemma\))d(in)i Fu(D)i Fx(suc)m(h)g(that)e (the)0 4567 y(diagram)1152 5190 y Fu(G)6 b Fx(\()p Ft(C)1332 5154 y Fr(0)1356 5190 y Ft(;)17 b Fx(-)o(\))298 b(Mor)1944 5205 y Fr(D)2005 5190 y Fx(\()p Fu(F)10 b Fx(\()p Ft(C)2240 5154 y Fr(0)2262 5190 y Fx(\))p Ft(;)17 b Fx(-)o(\))p 1498 5167 241 4 v 1656 5165 a Fl(-)1566 5230 y Fq(\030)1597 5249 y Fk(C)1644 5235 y Fh(0)1167 4702 y Fu(G)6 b Fx(\()p Ft(C)r(;)17 b Fx(-)o(\))323 b(Mor)1956 4717 y Fr(D)2016 4702 y Fx(\()p Fu(F)10 b Fx(\()p Ft(C)d Fx(\))p Ft(;)17 b Fx(-)o(\))p 1485 4679 267 4 v 1669 4677 a Fl(-)1576 4645 y Fq(\030)1607 4656 y Fk(C)p 1309 5097 4 351 v 1311 5097 a Fl(?)1082 4938 y Fr(G)t Fn(\()p Fq(f)s(;)p Fx(-)p Fn(\))p 2089 5097 V 2091 5097 a Fl(?)2130 4938 y Fn(Mor)o(\()p Fr(F)7 b Fn(\()p Fq(f)g Fn(\))p Fq(;)p Fx(-)q Fn(\))p eop %%Page: 42 42 42 41 bop 0 236 a Fo(42)1456 b(8.)77 b(TOOLBO)n(X)0 425 y Fx(comm)m(utes.)f(Because)44 b(of)f(the)h(uniqueness)h Fu(F)10 b Fx(\()p Ft(f)h Fx(\))42 b(and)i(b)s(ecause)g(of)f(the)h (functorialit)m(y)d(of)i Fu(G)0 541 y Fx(it)38 b(is)g(easy)i(to)f(see)h (that)e Fu(F)10 b Fx(\()p Ft(f)h(g)t Fx(\))37 b(=)h Fu(F)10 b Fx(\()p Ft(g)t Fx(\))p Fu(F)g Fx(\()p Ft(f)h Fx(\))37 b(and)i Fu(F)10 b Fx(\(1)2183 556 y Fq(C)2241 541 y Fx(\))38 b(=)g(1)2480 557 y Fr(F)7 b Fn(\()p Fq(C)e Fn(\))2690 541 y Fx(hold)38 b(and)h(that)g Fu(F)48 b Fx(is)38 b(a)0 658 y(con)m(tra)m(v)-5 b(arian)m(t)32 b(functor.)149 774 y(If)24 b Fu(F)320 738 y Fr(0)371 774 y Fx(:)k Fu(C)34 b(\000)-60 b(!)27 b(D)g Fx(is)c(giv)m(en)h(with)g Fu(G)1402 746 y(\030)1403 778 y Fx(=)1507 774 y(Mor)1683 789 y Fr(D)1744 774 y Fx(\()p Fu(F)1864 738 y Fr(0)1887 774 y Fx(-)o Ft(;)17 b Fx(-)o(\))24 b(then)h Ft(\036)i Fx(:)h(Mor)2587 789 y Fr(D)2648 774 y Fx(\()p Fu(F)10 b Fx(-)o Ft(;)17 b Fx(-)o(\))2942 746 y Fu(\030)2942 778 y Fx(=)3047 774 y(Mor)3223 789 y Fr(D)3284 774 y Fx(\()p Fu(F)3404 738 y Fr(0)3426 774 y Fx(-)p Ft(;)g Fx(-)o(\).)0 890 y(Hence)32 b(b)m(y)g(the)f(Y)-8 b(oneda)31 b(Lemma)e Ft( )t Fx(\()p Ft(C)7 b Fx(\))28 b(:)f Fu(F)10 b Fx(\()p Ft(C)d Fx(\))1843 862 y Fu(\030)1844 894 y Fx(=)1949 890 y Fu(F)2031 854 y Fr(0)2053 890 y Fx(\()p Ft(C)g Fx(\))31 b(is)f(an)h(isomorphism)d (for)i(all)f Ft(C)35 b Fu(2)28 b(C)6 b Fx(.)0 1006 y(With)32 b(these)i(isomorphisms)c(induced)k(b)m(y)f Ft(\036)f Fx(the)h(diagram)696 1732 y(Mor)872 1747 y Fr(D)933 1732 y Fx(\()p Fu(F)1053 1696 y Fr(0)1076 1732 y Fx(\()p Ft(C)1191 1696 y Fr(0)1214 1732 y Fx(\))p Ft(;)17 b Fx(-)o(\))902 b(Mor)2444 1747 y Fr(D)2505 1732 y Fx(\()p Fu(F)10 b Fx(\()p Ft(C)2740 1696 y Fr(0)2762 1732 y Fx(\))p Ft(;)17 b Fx(-)o(\))p 1395 1709 845 4 v 2157 1707 a Fl(-)1609 1777 y Fn(Mor\()p Fq( )r Fn(\()p Fq(C)1894 1754 y Fh(0)1918 1777 y Fn(\))p Fq(;)p Fx(-)o Fn(\))708 1245 y Fx(Mor)884 1260 y Fr(D)945 1245 y Fx(\()p Fu(F)1065 1208 y Fr(0)1088 1245 y Fx(\()p Ft(C)7 b Fx(\))p Ft(;)17 b Fx(-)o(\))925 b(Mor)2456 1260 y Fr(D)2516 1245 y Fx(\()p Fu(F)10 b Fx(\()p Ft(C)d Fx(\))p Ft(;)17 b Fx(-)o(\))p 1383 1221 868 4 v 2168 1219 a Fl(-)1621 1184 y Fn(Mor)o(\()p Fq( )r Fn(\()p Fq(C)5 b Fn(\))p Fq(;)p Fx(-)q Fn(\))p 1030 1639 4 351 v 1031 1639 a Fl(?)582 1481 y Fn(Mor\()p Fr(F)794 1457 y Fh(0)817 1481 y Fn(\()p Fq(f)i Fn(\))p Fq(;)p Fx(-)p Fn(\))p 2590 1639 V 2591 1639 a Fl(?)2630 1480 y Fn(Mor\()p Fr(F)g Fn(\()p Fq(f)g Fn(\))p Fq(;)p Fx(-)p Fn(\))0 1894 y Fx(comm)m(utes.)43 b(Hence)34 b(the)f(diagram)1402 2563 y Fu(F)10 b Fx(\()p Ft(C)d Fx(\))338 b Fu(F)2057 2527 y Fr(0)2080 2563 y Fx(\()p Ft(C)7 b Fx(\))p 1665 2540 281 4 v 1863 2538 a Fl(-)1726 2607 y Fq( )r Fn(\()p Fq(C)e Fn(\))1390 2076 y Fu(F)10 b Fx(\()p Ft(C)1587 2039 y Fr(0)1610 2076 y Fx(\))315 b Fu(F)2045 2039 y Fr(0)2068 2076 y Fx(\()p Ft(C)2183 2039 y Fr(0)2206 2076 y Fx(\))p 1677 2052 258 4 v 1852 2050 a Fl(-)1715 2015 y Fq( )r Fn(\()p Fq(C)1845 1992 y Fh(0)1868 2015 y Fn(\))p 1517 2470 4 351 v 1519 2470 a Fl(?)1304 2312 y Fr(F)1361 2288 y Fh(0)1384 2312 y Fn(\()p Fq(f)7 b Fn(\))p 2102 2470 V 2104 2470 a Fl(?)2143 2311 y Fr(F)g Fn(\()p Fq(f)g Fn(\))0 2749 y Fx(comm)m(utes.)43 b(Th)m(us)35 b Ft( )c Fx(:)d Fu(F)37 b(\000)-60 b(!)28 b(F)1229 2713 y Fr(0)1284 2749 y Fx(is)k(a)h(natural)e(isomorphism.)p 3529 2749 4 66 v 3533 2686 59 4 v 3533 2749 V 3591 2749 4 66 v 149 2925 a Fv(De\014nition)37 b(8.9.6.)49 b Fx(Let)38 b Fu(C)44 b Fx(and)38 b Fu(D)j Fx(b)s(e)d(categories)g(and)g Fu(F)46 b Fx(:)37 b Fu(C)44 b(\000)-60 b(!)37 b(D)j Fx(and)e Fu(G)43 b Fx(:)37 b Fu(D)j(\000)-60 b(!)37 b(C)0 3041 y Fx(b)s(e)d(co)m(v)-5 b(arian)m(t)32 b(functors.)47 b Fu(F)c Fx(is)32 b(called)h Fs(leftadjoint)f Fx(to)h Fu(G)40 b Fx(and)33 b Fu(G)40 b Fs(rightadjoint)32 b Fx(to)h Fu(F)43 b Fx(if)32 b(there)i(is)0 3157 y(a)g(natural)g (isomorphism)e(of)j(bifunctors)f Ft(\036)e Fx(:)f(Mor)1897 3172 y Fr(D)1958 3157 y Fx(\()p Fu(F)10 b Fx(-)o Ft(;)17 b Fx(-)o(\))31 b Fu(\000)-60 b(!)32 b Fx(Mor)2580 3172 y Fr(C)2625 3157 y Fx(\(-)p Ft(;)17 b Fu(G)6 b Fx(-)o(\))35 b(from)e Fu(C)3200 3121 y Fq(op)3298 3157 y Fu(\002)24 b(D)37 b Fx(to)0 3274 y Fv(Set)p Fx(.)149 3450 y Fv(Lemma)h(8.9.7.)49 b Fs(If)41 b Fu(F)50 b Fx(:)40 b Fu(C)46 b(\000)-58 b(!)41 b(D)j Fs(is)d(leftadjoint)g(to)h Fu(G)k Fx(:)41 b Fu(D)h(\000)-58 b(!)41 b(C)48 b Fs(then)41 b Fu(F)51 b Fs(is)42 b(uniquely)0 3566 y(determine)-5 b(d)36 b(by)h Fu(G)43 b Fs(up)37 b(to)g(isomorphism.)48 b(Similarly)36 b Fu(G)43 b Fs(is)37 b(uniquely)g(determine)-5 b(d)36 b(by)h Fu(F)46 b Fs(up)37 b(to)0 3682 y(isomorphism.)149 3858 y Fi(Pr)n(oof.)42 b Fx(No)m(w)f(w)m(e)g(pro)m(v)m(e)h(the)e(\014rst)h(claim.)63 b(Assume)41 b(that)f(also)f Fu(F)2763 3822 y Fr(0)2826 3858 y Fx(is)h(leftadjoin)m(t)e(to)i Fu(G)0 3975 y Fx(with)c Ft(\036)284 3939 y Fr(0)341 3975 y Fx(:)e(Mor)579 3990 y Fr(D)639 3975 y Fx(\()p Fu(F)759 3939 y Fr(0)782 3975 y Fx(-)o Ft(;)17 b Fx(-\))34 b Fu(\000)-60 b(!)34 b Fx(Mor)1290 3990 y Fr(C)1335 3975 y Fx(\(-)p Ft(;)17 b Fu(G)6 b Fx(-)o(\).)55 b(Then)37 b(w)m(e)h(ha)m(v)m(e)f(a)g(natural)e(isomorphism)f Ft(\036)3364 3939 y Fr(0)3387 3929 y(\000)p Fn(1)3481 3975 y Ft(\036)g Fx(:)0 4092 y(Mor)176 4107 y Fr(D)237 4092 y Fx(\()p Fu(F)10 b Fx(-)o Ft(;)17 b Fx(-)o(\))28 b Fu(\000)-60 b(!)27 b Fx(Mor)851 4107 y Fr(D)912 4092 y Fx(\()p Fu(F)1032 4056 y Fr(0)1055 4092 y Fx(-)o Ft(;)17 b Fx(-\).)43 b(By)33 b(Prop)s(osition)e(8.9.5)h(w)m(e)i(get)e Fu(F)2598 4064 y(\030)2599 4096 y Fx(=)2703 4092 y Fu(F)2785 4056 y Fr(0)2808 4092 y Fx(.)p 3529 4092 V 3533 4029 59 4 v 3533 4092 V 3591 4092 4 66 v 149 4268 a Fv(Lemma)38 b(8.9.8.)49 b Fs(A)43 b(functor)g Fu(G)48 b Fx(:)43 b Fu(D)i(\000)-58 b(!)42 b(C)49 b Fs(has)43 b(a)f(leftadjoint)g(functor)h (i\013)g(al)5 b(l)42 b(functors)0 4384 y Fx(Mor)176 4399 y Fr(C)221 4384 y Fx(\()p Ft(C)r(;)17 b Fu(G)6 b Fs(-)o Fx(\))35 b Fs(ar)-5 b(e)35 b(r)-5 b(epr)g(esentable.)149 4560 y Fi(Pr)n(oof.)42 b Fx(follo)m(ws)32 b(from)f(8.9.5.)p 3529 4560 V 3533 4497 59 4 v 3533 4560 V 3591 4560 4 66 v 149 4736 a Fv(Lemma)38 b(8.9.9.)49 b Fs(L)-5 b(et)35 b Fu(F)i Fx(:)28 b Fu(C)34 b(\000)-58 b(!)28 b(D)38 b Fs(and)c Fu(G)g Fx(:)28 b Fu(D)i(\000)-58 b(!)28 b(C)41 b Fs(b)-5 b(e)35 b(c)-5 b(ovariant)34 b(functors.)45 b(Then)551 4896 y Fx(Nat\(Id)838 4911 y Fr(C)883 4896 y Ft(;)17 b Fu(G)6 b(F)k Fx(\))27 b Fu(3)i Fx(\010)f Fu(7!)f(G)6 b Fs(-)p Fx(\010)p Fs(-)28 b Fu(2)g Fx(Nat\(Mor)2160 4911 y Fr(D)2221 4896 y Fx(\()p Fu(F)10 b Fs(-)o Ft(;)17 b Fs(-)p Fx(\))p Ft(;)g Fx(Mor)2711 4911 y Fr(C)2756 4896 y Fx(\()p Fs(-)p Ft(;)g Fu(G)6 b Fs(-)p Fx(\)\))0 5056 y Fs(is)35 b(a)f(bije)-5 b(ctive)34 b(map)h(with)f(inverse)g(map) 372 5216 y Fx(Nat\(Mor)746 5231 y Fr(D)807 5216 y Fx(\()p Fu(F)10 b Fs(-)o Ft(;)17 b Fs(-)p Fx(\))p Ft(;)g Fx(Mor)1297 5231 y Fr(C)1342 5216 y Fx(\()p Fs(-)p Ft(;)g Fu(G)6 b Fs(-)p Fx(\)\))27 b Fu(3)i Ft(\036)e Fu(7!)g Ft(\036)p Fx(\()p Fs(-)p Ft(;)17 b Fu(F)10 b Fs(-)o Fx(\)\(1)2385 5231 y Fr(F)d Fs(-)2481 5216 y Fx(\))27 b Fu(2)h Fx(Nat\(Id)2927 5231 y Fr(C)2972 5216 y Ft(;)17 b Fu(G)6 b(F)k Fx(\))p Ft(:)p eop %%Page: 43 43 43 42 bop 777 236 a Fo(9.)76 b(ADJOINT)33 b(FUNCTORS)f(AND)g(THE)i (YONED)n(A)d(LEMMA)675 b(43)0 425 y Fs(F)-7 b(urthermor)i(e)538 607 y Fx(Nat)o(\()p Fu(F)10 b(G)c Ft(;)17 b Fx(Id)1015 622 y Fr(C)1060 607 y Fx(\))28 b Fu(3)g Fx(\011)g Fu(7!)f Fx(\011)p Fs(-)p Fu(F)10 b Fs(-)27 b Fu(2)h Fx(Nat\(Mor)2174 622 y Fr(C)2219 607 y Fx(\()p Fs(-)o Ft(;)17 b Fu(G)6 b Fs(-)p Fx(\))p Ft(;)17 b Fx(Mor)2693 622 y Fr(D)2754 607 y Fx(\()p Fu(F)10 b Fs(-)o Ft(;)17 b Fs(-)o Fx(\)\))0 789 y Fs(is)35 b(a)f(bije)-5 b(ctive)34 b(map)h(with)f(inverse)g(map) 377 971 y Fx(Nat)o(\(Mor)751 986 y Fr(C)796 971 y Fx(\()p Fs(-)o Ft(;)17 b Fu(G)6 b Fs(-)p Fx(\))p Ft(;)17 b Fx(Mor)1270 986 y Fr(D)1331 971 y Fx(\()p Fu(F)10 b Fs(-)o Ft(;)17 b Fs(-)o Fx(\)\))28 b Fu(3)g Ft( )k Fu(7!)27 b Ft( )t Fx(\()p Fu(G)6 b Fs(-)p Ft(;)17 b Fs(-)p Fx(\)\(1)2392 986 y Fr(G)t Fs(-)2476 971 y Fx(\))28 b Fu(2)g Fx(Nat\()p Fu(F)10 b(G)c Ft(;)17 b Fx(Id)3113 986 y Fr(C)3158 971 y Fx(\))p Ft(:)149 1160 y Fi(Pr)n(oof.)42 b Fx(The)i(natural)e (transformation)e Fu(G)6 b Fx(-\010-)43 b(is)f(de\014ned)i(as)f(follo)m (ws.)73 b(Giv)m(en)43 b Ft(C)52 b Fu(2)46 b(C)6 b Fx(,)0 1276 y Ft(D)53 b Fu(2)e(D)d Fx(and)e Ft(f)61 b Fu(2)51 b Fx(Mor)981 1291 y Fr(D)1042 1276 y Fx(\()p Fu(F)10 b Fx(\()p Ft(C)d Fx(\))p Ft(;)17 b(D)s Fx(\))44 b(then)j(let)e(\()p Fu(G)6 b Fx(-\010-\)\()p Ft(C)r(;)17 b(D)s Fx(\)\()p Ft(f)11 b Fx(\))48 b(:=)j Fu(G)6 b Fx(\()p Ft(f)11 b Fx(\)\010\()p Ft(C)c Fx(\))50 b(:)h Ft(C)57 b Fu(\000)-60 b(!)0 1392 y(G)6 b(F)k Fx(\()p Ft(C)d Fx(\))27 b Fu(\000)-60 b(!)28 b(G)6 b Fx(\()p Ft(D)s Fx(\).)43 b(It)33 b(is)f(easy)h(to)g(c)m (hec)m(k)i(the)e(prop)s(erties)f(of)g(a)g(natural)g(transformation.)149 1509 y(Giv)m(en)51 b(\010)h(then)g(one)f(obtains)f(b)m(y)i(comp)s (osition)d(of)i(the)g(t)m(w)m(o)h(maps)f Fu(G)6 b Fx(\(1)3033 1524 y Fr(F)h Fn(\()p Fq(C)e Fn(\))3204 1509 y Fx(\)\010\()p Ft(C)i Fx(\))59 b(=)0 1625 y Fu(G)6 b(F)k Fx(\(1)234 1640 y Fq(C)293 1625 y Fx(\)\010\()p Ft(C)d Fx(\))27 b(=)h(\010\()p Ft(C)7 b Fx(\).)44 b(Giv)m(en)32 b Ft(\036)g Fx(one)h(obtains)531 1805 y Fu(G)6 b Fx(\()p Ft(f)11 b Fx(\)\()p Ft(\036)p Fx(\()p Ft(C)r(;)17 b Fu(F)10 b Fx(\()p Ft(C)d Fx(\)\)\(1)1341 1821 y Fr(F)g Fn(\()p Fq(C)e Fn(\))1509 1805 y Fx(\))28 b(=)f(Mor)1855 1820 y Fr(C)1900 1805 y Fx(\()p Ft(C)r(;)17 b Fu(G)6 b Fx(\()p Ft(f)11 b Fx(\)\))p Ft(\036)p Fx(\()p Ft(C)r(;)17 b Fu(F)10 b Fx(\()p Ft(C)d Fx(\)\)\(1)2864 1821 y Fr(F)g Fn(\()p Fq(C)e Fn(\))3031 1805 y Fx(\))531 1921 y(=)27 b Ft(\036)p Fx(\()p Ft(C)r(;)17 b(D)s Fx(\)Mor)1143 1936 y Fr(D)1204 1921 y Fx(\()p Fu(F)10 b Fx(\()p Ft(C)d Fx(\))p Ft(;)17 b(f)11 b Fx(\)\(1)1705 1937 y Fr(F)c Fn(\()p Fq(C)e Fn(\))1874 1921 y Fx(\))28 b(=)f Ft(\036)p Fx(\()p Ft(C)r(;)17 b(D)s Fx(\)\()p Ft(f)11 b Fx(\))p Ft(:)0 2100 y Fx(The)34 b(second)f(part)g (of)f(the)h(lemma)d(is)i(pro)m(v)m(ed)i(similarly)-8 b(.)p 3529 2100 4 66 v 3533 2037 59 4 v 3533 2100 V 3591 2100 4 66 v 149 2288 a Fv(Prop)s(osition)36 b(8.9.10.)49 b Fs(L)-5 b(et)285 2470 y Ft(\036)28 b Fx(:)g(Mor)602 2485 y Fr(D)662 2470 y Fx(\()p Fu(F)10 b Fs(-)p Ft(;)17 b Fs(-)o Fx(\))28 b Fu(\000)-58 b(!)28 b Fx(Mor)1284 2485 y Fr(C)1329 2470 y Fx(\()p Fs(-)p Ft(;)17 b Fu(G)6 b Fs(-)p Fx(\))134 b Fs(and)g Ft( )32 b Fx(:)c(Mor)2333 2485 y Fr(C)2378 2470 y Fx(\()p Fs(-)o Ft(;)17 b Fu(G)6 b Fs(-)p Fx(\))28 b Fu(\000)-58 b(!)28 b Fx(Mor)2983 2485 y Fr(D)3044 2470 y Fx(\()p Fu(F)10 b Fs(-)o Ft(;)17 b Fs(-)p Fx(\))0 2652 y Fs(b)-5 b(e)31 b(natur)-5 b(al)32 b(tr)-5 b(ansformations)31 b(with)h(asso)-5 b(ciate)g(d)30 b(natur)-5 b(al)32 b(tr)-5 b(ansformations)31 b(\(by)h(L)-5 b(emma)31 b(8.9.9\))0 2768 y Fx(\010)d(:)g(Id)242 2783 y Fr(C)315 2768 y Fu(\000)-58 b(!)28 b(G)6 b(F)45 b Fs(r)-5 b(esp.)44 b Fx(\011)28 b(:)f Fu(F)10 b(G)34 b(\000)-58 b(!)28 b Fx(Id)1458 2783 y Fr(D)1519 2768 y Fs(.)149 2908 y(1\))35 b(Then)f(we)g(have)h Ft(\036 )c Fx(=)d(id)1234 2923 y Fn(Mor)o(\()p Fs(-)q Fq(;)p Fr(G)t Fs(-)o Fn(\))1590 2908 y Fs(if)35 b(and)f(only)h(if)g Fx(\()p Fu(G)2343 2851 y Fn(\010)p Fr(G)2311 2908 y Fu(\000)-16 b(!)28 b(G)6 b(F)k(G)2769 2851 y Fr(G)t Fn(\011)2739 2908 y Fu(\000)-16 b(!)27 b(G)6 b Fx(\))28 b(=)g(id)3243 2923 y Fr(G)3293 2908 y Fs(.)149 3060 y(2\))35 b(We)g(also)f(have)g Ft( )t(\036)28 b Fx(=)f(id)1212 3076 y Fn(Mor)o(\()p Fr(F)7 b Fs(-)p Fq(;)p Fs(-)p Fn(\))1579 3060 y Fs(if)35 b(and)f(only)h(if)g Fx(\()p Fu(F)2343 3004 y Fr(F)7 b Fn(\010)2317 3060 y Fu(\000)-16 b(!)27 b(F)10 b(G)c(F)2785 3004 y Fn(\011)p Fr(F)2761 3060 y Fu(\000)-16 b(!)27 b(F)10 b Fx(\))27 b(=)h(id)3281 3075 y Fr(F)3342 3060 y Fs(.)149 3249 y Fi(Pr)n(oof.)42 b Fx(W)-8 b(e)33 b(get)759 3424 y Fu(G)6 b Fx(\011\()p Ft(D)s Fx(\)\010)p Fu(G)g Fx(\()p Ft(D)s Fx(\))27 b(=)h Fu(G)6 b Fx(\011\()p Ft(D)s Fx(\))p Ft(\036)p Fx(\()p Fu(G)g Fx(\()p Ft(D)s Fx(\))p Ft(;)17 b Fu(F)10 b(G)c Fx(\()p Ft(D)s Fx(\)\)\(1)2584 3439 y Fr(F)h(G)t Fn(\()p Fq(D)r Fn(\))2803 3424 y Fx(\))759 3540 y(=)27 b(Mor)1039 3555 y Fr(C)1084 3540 y Fx(\()p Fu(G)6 b Fx(\()p Ft(D)s Fx(\))p Ft(;)17 b Fu(G)6 b Fx(\011\()p Ft(D)s Fx(\)\))p Ft(\036)p Fx(\()p Fu(G)g Fx(\()p Ft(D)s Fx(\))p Ft(;)17 b Fu(F)10 b(G)c Fx(\()p Ft(D)s Fx(\)\)\(1)2527 3555 y Fr(F)h(G)t Fn(\()p Fq(D)r Fn(\))2745 3540 y Fx(\))759 3656 y(=)27 b Ft(\036)p Fx(\()p Fu(G)6 b Fx(\()p Ft(D)s Fx(\))p Ft(;)17 b(D)s Fx(\)Mor)1524 3671 y Fr(D)1585 3656 y Fx(\()p Fu(F)10 b(G)c Fx(\()p Ft(D)s Fx(\))p Ft(;)17 b Fx(\011\()p Ft(D)s Fx(\)\)\(1)2335 3672 y Fr(F)7 b(G)t Fn(\()p Fq(D)r Fn(\))2555 3656 y Fx(\))759 3772 y(=)27 b Ft(\036)p Fx(\()p Fu(G)6 b Fx(\()p Ft(D)s Fx(\))p Ft(;)17 b(D)s Fx(\)\(\011\()p Ft(D)s Fx(\)\))759 3889 y(=)27 b Ft(\036)p Fx(\()p Fu(G)6 b Fx(\()p Ft(D)s Fx(\))p Ft(;)17 b(D)s Fx(\))p Ft( )t Fx(\()p Fu(G)6 b Fx(\()p Ft(D)s Fx(\))p Ft(;)17 b(D)s Fx(\)\(1)1932 3904 y Fr(G)t Fn(\()p Fq(D)r Fn(\))2095 3889 y Fx(\))759 4005 y(=)27 b Ft(\036 )t Fx(\()p Fu(G)6 b Fx(\()p Ft(D)s Fx(\))p Ft(;)17 b(D)s Fx(\)\(1)1503 4020 y Fr(G)t Fn(\()p Fq(D)r Fn(\))1667 4005 y Fx(\))p Ft(:)0 4183 y Fx(Similarly)29 b(w)m(e)k(get)383 4358 y Ft(\036 )t Fx(\()p Ft(C)r(;)17 b(D)s Fx(\)\()p Ft(f)11 b Fx(\))26 b(=)h Ft(\036)p Fx(\()p Ft(C)r(;)17 b(D)s Fx(\))p Ft( )t Fx(\()p Ft(C)r(;)g(D)s Fx(\)\()p Ft(f)11 b Fx(\))25 b(=)i Fu(G)6 b Fx(\(\011\()p Ft(D)s Fx(\))p Fu(F)k Fx(\()p Ft(f)h Fx(\)\)\010\()p Ft(C)c Fx(\))383 4474 y(=)27 b Fu(G)6 b Fx(\011\()p Ft(D)s Fx(\))p Fu(G)g(F)k Fx(\()p Ft(f)h Fx(\)\010\()p Ft(C)c Fx(\))27 b(=)h Fu(G)6 b Fx(\011\()p Ft(D)s Fx(\)\010)p Fu(G)g Fx(\()p Ft(D)s Fx(\))p Ft(f)f(:)p 3147 4474 V 3151 4411 59 4 v 3151 4474 V 3209 4474 4 66 v 149 4659 a Fv(Corollary)36 b(8.9.11.)50 b Fs(L)-5 b(et)36 b Fu(F)k Fx(:)30 b Fu(C)37 b(\000)-58 b(!)30 b(D)39 b Fs(and)d Fu(G)g Fx(:)30 b Fu(D)j(\000)-58 b(!)31 b(C)42 b Fs(b)-5 b(e)36 b(functors.)49 b Fu(F)d Fs(is)36 b(leftadjoint)0 4775 y(to)25 b Fu(G)31 b Fs(if)25 b(and)f(only)h(if)g(ther)-5 b(e)25 b(ar)-5 b(e)25 b(natur)-5 b(al)25 b(tr)-5 b(ansformations)24 b Fx(\010)k(:)g(Id)2399 4790 y Fr(C)2472 4775 y Fu(\000)-58 b(!)28 b(G)6 b(F)35 b Fs(and)24 b Fx(\011)k(:)f Fu(F)10 b(G)34 b(\000)-58 b(!)28 b Fx(Id)3539 4790 y Fr(D)0 4891 y Fs(such)35 b(that)g Fx(\()p Fu(G)6 b Fx(\011\)\(\010)p Fu(G)g Fx(\))28 b(=)g(id)1062 4906 y Fr(G)1147 4891 y Fs(and)34 b Fx(\(\011)p Fu(F)10 b Fx(\)\()p Fu(F)g Fx(\010\))27 b(=)g(id)2009 4906 y Fr(F)2071 4891 y Fs(.)149 5080 y Fv(De\014nition)37 b(8.9.12.)49 b Fx(The)36 b(natural)d (transformations)g(\010)f(:)g(Id)2540 5095 y Fr(C)2617 5080 y Fu(\000)-60 b(!)31 b(G)6 b(F)45 b Fx(and)35 b(\011)c(:)h Fu(F)10 b(G)37 b(\000)-60 b(!)0 5196 y Fx(Id)89 5211 y Fr(D)181 5196 y Fx(giv)m(en)32 b(in)e(8.9.11)h(are)g(called)f Fs(unit)i Fx(and)f Fs(c)-5 b(ounit)31 b Fx(resp.)44 b(for)31 b(the)h(adjoin)m(t)e(functors)i Fu(F)40 b Fx(and)32 b Fu(G)6 b Fx(.)p eop %%Page: 44 44 44 43 bop 0 236 a Fo(44)1456 b(8.)77 b(TOOLBO)n(X)149 425 y Fv(Problem)36 b(8.9.16.)49 b Fx(1.)58 b(Let)1283 440 y Fq(R)1341 425 y Ft(M)1435 440 y Fq(S)1523 425 y Fx(b)s(e)38 b(a)f(bimo)s(dule.)56 b(Sho)m(w)39 b(that)e(the)h(functor)f Ft(M)g Fu(\012)3428 440 y Fq(S)3504 425 y Fx(-)f(:)0 556 y Fq(S)51 541 y Fu(M)g(\000)-60 b(!)361 556 y Fq(R)418 541 y Fu(M)38 b Fx(is)f(leftadjoin)m(t)f(to)h(Hom)1481 556 y Fq(R)1538 541 y Fx(\()p Ft(M)5 b(;)17 b Fx(-\))36 b(:)1890 556 y Fq(R)1948 541 y Fu(M)g(\000)-60 b(!)2258 556 y Fq(S)2308 541 y Fu(M)p Fx(.)59 b(Determine)37 b(the)h(asso)s (ciated)0 658 y(unit)32 b(and)h(counit.)149 774 y(b\))22 b(Sho)m(w)h(that)e(there)i(is)e(a)h(natural)e(isomorphism)f(Map)q(\()p Ft(A)p Fu(\002)p Ft(B)5 b(;)17 b(C)7 b Fx(\))2643 746 y Fu(\030)2643 778 y Fx(=)2748 774 y(Map\()p Ft(B)e(;)17 b Fx(Map\()p Ft(A;)g(C)7 b Fx(\)\).)0 890 y(Determine)32 b(the)h(asso)s(ciated)f(unit)g(and)h(counit.)149 1006 y(c\))49 b(Sho)m(w)h(that)e(there)h(is)g(a)f(natural)f(isomorphism)f Fp(K)18 b Fx(-)6 b Fv(Alg)n Fx(\()p Fp(K)18 b Ft(G;)f(A)p Fx(\))2884 979 y Fu(\030)2885 1011 y Fx(=)3017 1006 y Fv(Gr)p Fx(\()p Ft(G;)g(U)10 b Fx(\()p Ft(A)p Fx(\)\).)0 1123 y(Determine)32 b(the)h(asso)s(ciated)f(unit)g(and)h(counit.)149 1239 y(d\))39 b(Sho)m(w)h(that)f(there)g(is)g(a)f(natural)g (isomorphism)e Fp(K)18 b Fx(-)5 b Fv(Alg)o Fx(\()p Ft(U)10 b Fx(\()p Fm(g)p Fx(\))p Ft(;)17 b(A)p Fx(\))2849 1211 y Fu(\030)2850 1243 y Fx(=)2965 1239 y(Lie-Alg)n(\()p Fm(g)p Ft(;)g(A)3483 1203 y Fq(L)3535 1239 y Fx(\))p Ft(:)0 1355 y Fx(Determine)32 b(the)h(corresp)s(onding)f(leftadjoin)m (t)f(functor)i(and)f(the)h(asso)s(ciated)g(unit)f(and)g(counit.)149 1539 y Fv(De\014nition)37 b(8.9.13.)49 b Fx(Let)39 b Fu(G)45 b Fx(:)38 b Fu(D)j(\000)-60 b(!)39 b(C)45 b Fx(b)s(e)39 b(a)g(co)m(v)-5 b(arian)m(t)38 b(functor.)63 b Fu(G)45 b Fx(generates)40 b(a)f Fs(\(c)-5 b(o-)0 1656 y(\)universal)34 b(pr)-5 b(oblem)32 b Fx(a)g(follo)m(ws:)149 1772 y(Giv)m(en)38 b Ft(C)43 b Fu(2)37 b(C)6 b Fx(.)58 b(Find)37 b(an)g(ob)5 b(ject)39 b Fu(F)10 b Fx(\()p Ft(C)d Fx(\))35 b Fu(2)i(D)j Fx(and)e(a)f(morphism)f Ft(\023)g Fx(:)h Ft(C)43 b Fu(\000)-60 b(!)36 b(G)6 b Fx(\()p Fu(F)k Fx(\()p Ft(C)d Fx(\)\))37 b(in)0 1888 y Fu(C)k Fx(suc)m(h)36 b(that)e(there)h(is)f(a)g(unique)h (morphism)e Ft(g)h Fx(:)d Fu(F)10 b Fx(\()p Ft(C)d Fx(\))30 b Fu(\000)-60 b(!)31 b Ft(D)37 b Fx(in)d Fu(D)j Fx(for)d(eac)m(h)h(ob)5 b(ject)35 b Ft(D)f Fu(2)d(D)0 2004 y Fx(and)i(for)f(eac)m(h)h(morphism) e Ft(f)38 b Fx(:)28 b Ft(C)35 b Fu(\000)-60 b(!)27 b(G)6 b Fx(\()p Ft(D)s Fx(\))33 b(in)f Fu(C)38 b Fx(suc)m(h)c(that)f(the)g (diagram)1443 2171 y Ft(C)268 b Fu(G)6 b Fx(\()p Fu(F)k Fx(\()p Ft(C)d Fx(\)\))p 1549 2147 204 4 v 1670 2145 a Fl(-)1638 2127 y Fq(\023)1624 2406 y(f)1550 2297 y Fl(@)1633 2380 y(@)1716 2463 y(@)1799 2546 y(@)1818 2565 y(@)-83 b(R)1857 2658 y Fu(G)6 b Fx(\()p Ft(D)s Fx(\))p 1967 2565 4 351 v 1969 2565 a Fl(?)2008 2406 y Fr(G)t Fn(\()p Fq(g)r Fn(\))0 2811 y Fx(comm)m(utes.)149 2927 y(A)36 b(pair)e(\()p Fu(F)10 b Fx(\()p Ft(C)d Fx(\))p Ft(;)17 b(\023)p Fx(\))35 b(that)h(satis\014es)g(the)g(ab)s(o)m(v)m(e)g (conditions)f(is)g(called)f(a)h Fs(universal)i(solution)0 3043 y Fx(of)32 b(the)h(\(co-\)univ)m(ersal)f(problem)f(de\014ned)j(b)m (y)g Fu(G)39 b Fx(and)32 b Ft(C)7 b Fx(.)149 3159 y(Let)30 b Fu(F)37 b Fx(:)27 b Fu(C)34 b(\000)-60 b(!)28 b(D)k Fx(b)s(e)d(a)f(co)m(v)-5 b(arian)m(t)29 b(functor.)42 b Fu(F)c Fx(generates)30 b(a)f Fs(universal)i(pr)-5 b(oblem)28 b Fx(a)g(follo)m(ws:)149 3276 y(Giv)m(en)j Ft(D)g Fu(2)d(D)s Fx(.)43 b(Find)30 b(an)h(ob)5 b(ject)32 b Fu(G)6 b Fx(\()p Ft(D)s Fx(\))27 b Fu(2)h(C)37 b Fx(and)32 b(a)f(morphism)e Ft(\027)34 b Fx(:)28 b Fu(F)10 b Fx(\()p Fu(G)c Fx(\()p Ft(D)s Fx(\)\))27 b Fu(\000)-60 b(!)27 b Ft(D)34 b Fx(in)c Fu(D)0 3392 y Fx(suc)m(h)35 b(that)d(there)i(is)f(a)g(unique)g (morphism)f Ft(g)f Fx(:)e Ft(C)36 b Fu(\000)-60 b(!)28 b(G)6 b Fx(\()p Ft(D)s Fx(\))33 b(in)f Fu(C)40 b Fx(for)32 b(eac)m(h)i(ob)5 b(ject)34 b Ft(C)i Fu(2)29 b(C)39 b Fx(and)0 3508 y(for)32 b(eac)m(h)i(morphism)c Ft(f)39 b Fx(:)27 b Fu(F)10 b Fx(\()p Ft(C)d Fx(\))27 b Fu(\000)-60 b(!)28 b Ft(D)35 b Fx(in)d Fu(D)j Fx(suc)m(h)f(that)f(the)g(diagram) 1476 4162 y Fu(F)10 b(G)c Fx(\()p Ft(D)s Fx(\))292 b Ft(D)p 1811 4139 235 4 v 1963 4137 a Fl(-)1909 4184 y Fq(\027)1933 3910 y(f)1697 3801 y Fl(@)1780 3884 y(@)1863 3967 y(@)1946 4050 y(@)1965 4068 y(@)-83 b(R)1512 3674 y Fu(F)10 b Fx(\()p Ft(C)d Fx(\))p 1627 4069 4 351 v 1629 4069 a Fl(?)1442 3910 y Fr(F)g Fn(\()p Fq(g)r Fn(\))0 4314 y Fx(comm)m(utes.)149 4431 y(A)35 b(pair)f(\()p Fu(G)6 b Fx(\()p Ft(D)s Fx(\))p Ft(;)17 b(\027)6 b Fx(\))34 b(that)h(satis\014es)g(the)g(ab)s(o)m(v)m(e)g(conditions)f(is)g(called) g(a)g Fs(universal)i(solution)0 4547 y Fx(of)c(the)h(\(co-\)univ)m (ersal)f(problem)f(de\014ned)j(b)m(y)g Fu(F)42 b Fx(and)33 b Ft(D)s Fx(.)149 4731 y Fv(Prop)s(osition)j(8.9.14.)49 b Fs(L)-5 b(et)39 b Fu(F)k Fx(:)33 b Fu(C)40 b(\000)-58 b(!)34 b(D)41 b Fs(b)-5 b(e)38 b(leftadjoint)f(to)h Fu(G)i Fx(:)34 b Fu(D)i(\000)-58 b(!)34 b(C)6 b Fs(.)54 b(Then)38 b Fu(F)10 b Fx(\()p Ft(C)d Fx(\))0 4848 y Fs(and)44 b(the)g(unit)g Ft(\023)h Fx(=)g(\010\()p Ft(C)7 b Fx(\))45 b(:)g Ft(C)52 b Fu(\000)-58 b(!)45 b(G)6 b(F)k Fx(\()p Ft(C)d Fx(\))44 b Fs(form)f(a)h(\(c)-5 b(o-\)universal)43 b(solution)h(for)f(the)i(\(c) -5 b(o-)0 4964 y(\)universal)34 b(pr)-5 b(oblem)34 b(de\014ne)-5 b(d)34 b(by)h Fu(G)41 b Fs(and)34 b Ft(C)7 b Fs(.)149 5080 y(F)-7 b(urthermor)i(e)44 b Fu(G)6 b Fx(\()p Ft(D)s Fx(\))44 b Fs(and)f(the)h(c)-5 b(ounit)44 b Ft(\027)52 b Fx(=)45 b(\011\()p Ft(D)s Fx(\))f(:)h Fu(F)10 b(G)c Fx(\()p Ft(D)s Fx(\))44 b Fu(\000)-58 b(!)45 b Ft(D)i Fs(form)d(a)g(universal)0 5196 y(solution)35 b(for)f(the)h(universal)f (pr)-5 b(oblem)34 b(de\014ne)-5 b(d)34 b(by)h Fu(F)44 b Fs(and)35 b Ft(D)s Fs(.)p eop %%Page: 45 45 45 44 bop 777 236 a Fo(9.)76 b(ADJOINT)33 b(FUNCTORS)f(AND)g(THE)i (YONED)n(A)d(LEMMA)675 b(45)149 425 y Fi(Pr)n(oof.)42 b Fx(By)e(Theorem)f(8.9.10)f(the)h(morphisms)f Ft(\036)g Fx(:)g(Mor)2413 440 y Fr(D)2473 425 y Fx(\()p Fu(F)10 b Fx(-)o Ft(;)17 b Fx(-)o(\))39 b Fu(\000)-60 b(!)38 b Fx(Mor)3109 440 y Fr(C)3154 425 y Fx(\(-)p Ft(;)17 b Fu(G)6 b Fx(-)o(\))39 b(and)0 541 y Ft( )32 b Fx(:)27 b(Mor)326 556 y Fr(C)371 541 y Fx(\(-)o Ft(;)17 b Fu(G)6 b Fx(-)o(\))28 b Fu(\000)-60 b(!)28 b Fx(Mor)969 556 y Fr(D)1030 541 y Fx(\()p Fu(F)10 b Fx(-)o Ft(;)17 b Fx(-)o(\))30 b(are)g(in)m(v)m(erses)i(of)e(eac)m(h)h(other.)43 b(They)32 b(are)e(de\014ned)i(with)e(unit)0 658 y(and)43 b(counit)e(as)i Ft(\036)p Fx(\()p Ft(C)r(;)17 b(D)s Fx(\)\()p Ft(g)t Fx(\))42 b(=)i Fu(G)6 b Fx(\()p Ft(g)t Fx(\)\010\()p Ft(C)h Fx(\))42 b(resp.)74 b Ft( )t Fx(\()p Ft(C)r(;)17 b(D)s Fx(\)\()p Ft(f)11 b Fx(\))43 b(=)h(\011\()p Ft(D)s Fx(\))p Fu(F)10 b Fx(\()p Ft(f)h Fx(\).)72 b(Hence)43 b(for)0 774 y(eac)m(h)c Ft(f)47 b Fx(:)37 b Ft(C)44 b Fu(\000)-60 b(!)37 b(G)6 b Fx(\()p Ft(D)s Fx(\))37 b(there)i(is)e(a)h (unique)g Ft(g)i Fx(:)d Fu(F)10 b Fx(\()p Ft(C)d Fx(\))36 b Fu(\000)-60 b(!)37 b Ft(D)k Fx(suc)m(h)e(that)f Ft(f)47 b Fx(=)37 b Ft(\036)p Fx(\()p Ft(C)r(;)17 b(D)s Fx(\)\()p Ft(g)t Fx(\))34 b(=)0 890 y Fu(G)6 b Fx(\()p Ft(g)t Fx(\)\010\()p Ft(C)h Fx(\))27 b(=)h Fu(G)6 b Fx(\()p Ft(g)t Fx(\))p Ft(\023)p Fx(.)149 1006 y(The)34 b(second)g(statemen)m(t)f(follo)m(ws)e (analogously)-8 b(.)p 3529 1006 4 66 v 3533 944 59 4 v 3533 1006 V 3591 1006 4 66 v 149 1169 a Fv(Remark)37 b(8.9.15.)50 b Fx(If)43 b Fu(G)53 b Fx(:)47 b Fu(D)j(\000)-60 b(!)46 b(C)k Fx(and)44 b Ft(C)54 b Fu(2)47 b(C)j Fx(are)44 b(giv)m(en)g(then)g(the)h(\(co-\)univ)m(ersal)0 1285 y(solution)33 b(\()p Fu(F)10 b Fx(\()p Ft(C)d Fx(\))p Ft(;)17 b(\023)32 b Fx(:)g Ft(C)38 b Fu(\000)-60 b(!)32 b(G)6 b Fx(\()p Ft(D)s Fx(\)\))35 b(can)g(b)s(e)g(considered)h(as)f (the)h(b)s(est)f(\(co-\)appro)m(ximation)d(of)0 1401 y(the)42 b(ob)5 b(ject)42 b Ft(C)48 b Fx(in)40 b Fu(C)48 b Fx(b)m(y)42 b(an)f(ob)5 b(ject)42 b Ft(D)i Fx(in)d Fu(D)i Fx(with)e(the)h(help)f(of)g(a)g(functor)g Fu(G)6 b Fx(.)70 b(The)42 b(ob)5 b(ject)0 1517 y Ft(D)30 b Fu(2)e(D)36 b Fx(turns)d(out)f(to)h(b)s(e)f Fu(F)10 b Fx(\()p Ft(C)d Fx(\).)149 1633 y(If)48 b Fu(F)63 b Fx(:)54 b Fu(C)60 b(\000)-60 b(!)54 b(D)c Fx(and)e Ft(D)56 b Fu(2)e(D)d Fx(are)d(giv)m(en)g(then)g(the)g(univ)m(ersal)g(solution)e(\()p Fu(G)6 b Fx(\()p Ft(D)s Fx(\))p Ft(;)17 b(\027)60 b Fx(:)0 1750 y Fu(F)10 b(G)c Fx(\()p Ft(D)s Fx(\))28 b Fu(\000)-60 b(!)29 b Ft(D)s Fx(\))k(can)g(b)s(e)h(considered)g(as)f(the)h(b)s(est)g (appro)m(ximation)d(of)h(the)i(ob)5 b(ject)34 b Ft(D)i Fx(in)c Fu(D)k Fx(b)m(y)0 1866 y(an)i(ob)5 b(ject)40 b Ft(C)45 b Fx(in)38 b Fu(C)44 b Fx(with)38 b(the)h(help)f(of)g(a)g (functor)h Fu(F)10 b Fx(.)60 b(The)40 b(ob)5 b(ject)39 b Ft(C)45 b Fu(2)38 b(C)44 b Fx(turns)39 b(out)g(to)f(b)s(e)0 1982 y Fu(G)6 b Fx(\()p Ft(D)s Fx(\).)149 2144 y Fv(Prop)s(osition)36 b(8.9.16.)49 b Fs(Given)39 b Fu(G)k Fx(:)36 b Fu(D)i(\000)-58 b(!)37 b(C)6 b Fs(.)59 b(Assume)39 b(that)h(for)f(e)-5 b(ach)39 b Ft(C)k Fu(2)36 b(C)46 b Fs(the)39 b(uni-)0 2261 y(versal)k(pr)-5 b(oblem)43 b(de\014ne)-5 b(d)42 b(by)i Fu(G)50 b Fs(and)43 b Ft(C)50 b Fs(is)44 b(solvable.)70 b(Then)42 b(ther)-5 b(e)44 b(is)f(a)h(leftadjoint)f(functor)0 2377 y Fu(F)37 b Fx(:)28 b Fu(C)34 b(\000)-58 b(!)28 b(D)38 b Fs(to)d Fu(G)6 b Fs(.)149 2493 y(Given)34 b Fu(F)j Fx(:)28 b Fu(C)34 b(\000)-58 b(!)28 b(D)s Fs(.)44 b(Assume)34 b(that)h(for)f(e)-5 b(ach)33 b Ft(D)e Fu(2)d(D)36 b Fs(the)e(universal)g(pr)-5 b(oblem)33 b(de\014ne)-5 b(d)33 b(by)0 2609 y Fu(F)44 b Fs(and)35 b Ft(D)i Fs(is)e(solvable.)43 b(Then)34 b(ther)-5 b(e)35 b(is)g(a)g(leftadjoint)f(functor)h Fu(G)f Fx(:)27 b Fu(D)k(\000)-58 b(!)28 b(C)41 b Fs(to)35 b Fu(F)10 b Fs(.)149 2771 y Fi(Pr)n(oof.)42 b Fx(Assume)h(that)f(the)h (\(co-\)univ)m(ersal)e(problem)g(de\014ned)j(b)m(y)f Fu(G)48 b Fx(and)43 b Ft(C)49 b Fx(is)41 b(solv)m(ed)0 2888 y(b)m(y)d Ft(\023)e Fx(:)f Ft(C)43 b Fu(\000)-60 b(!)35 b(F)10 b Fx(\()p Ft(C)d Fx(\).)56 b(Then)39 b(the)e(map)f(Mor) 1684 2903 y Fr(C)1729 2888 y Fx(\()p Ft(C)r(;)17 b Fu(G)6 b Fx(\()p Ft(D)s Fx(\)\))35 b Fu(3)g Ft(f)47 b Fu(7!)35 b Ft(g)j Fu(2)e Fx(Mor)2875 2903 y Fr(D)2936 2888 y Fx(\()p Fu(F)10 b Fx(\()p Ft(C)d Fx(\))p Ft(;)17 b(D)s Fx(\))35 b(with)0 3004 y Fu(G)6 b Fx(\()p Ft(g)t Fx(\))p Ft(\023)41 b Fx(=)g Ft(f)51 b Fx(is)40 b(bijectiv)m(e.)67 b(The)41 b(in)m(v)m(erse)h(map)e(is)g(giv)m(en)g(b)m(y)i Ft(g)i Fu(7!)d(G)6 b Fx(\()p Ft(g)t Fx(\))p Ft(\023)p Fx(.)66 b(This)41 b(is)f(a)g(natural)0 3120 y(transformation)30 b(since)j(the)g(diagram)686 3787 y(Mor)862 3802 y Fr(D)923 3787 y Fx(\()p Fu(F)10 b Fx(\()p Ft(C)d Fx(\))p Ft(;)17 b(D)1324 3751 y Fr(0)1346 3787 y Fx(\))880 b(Mor)2441 3802 y Fr(C)2486 3787 y Fx(\()p Ft(C)r(;)17 b Fu(G)6 b Fx(\()p Ft(D)2827 3751 y Fr(0)2849 3787 y Fx(\)\))p 1412 3764 824 4 v 2153 3762 a Fl(-)1744 3831 y Fr(G)t Fn(\()p Fx(-)p Fn(\))p Fq(\023)697 3299 y Fx(Mor)874 3314 y Fr(D)934 3299 y Fx(\()p Fu(F)k Fx(\()p Ft(C)d Fx(\))p Ft(;)17 b(D)s Fx(\))903 b(Mor)2452 3314 y Fr(C)2497 3299 y Fx(\()p Ft(C)r(;)17 b Fu(G)6 b Fx(\()p Ft(D)s Fx(\)\))p 1400 3276 848 4 v 2165 3274 a Fl(-)1744 3238 y Fr(G)t Fn(\()p Fx(-)p Fn(\))p Fq(\023)p 1033 3694 4 351 v 1035 3694 a Fl(?)533 3535 y Fn(Mor)660 3546 y Fh(D)714 3535 y Fn(\()p Fr(F)h Fn(\()p Fq(C)e Fn(\))p Fq(;h)p Fn(\))p 2593 3694 V 2595 3694 a Fl(?)2634 3535 y Fn(Mor)2761 3546 y Fh(C)2801 3535 y Fn(\()p Fq(C)q(;)p Fr(G)t Fn(\()p Fq(h)p Fn(\)\))0 3971 y Fx(comm)m(utes)32 b(for)h(eac)m(h)g Ft(h)28 b Fu(2)g Fx(Mor)1181 3986 y Fq(D)1245 3971 y Fx(\()p Ft(D)s(;)17 b(D)1495 3935 y Fr(0)1517 3971 y Fx(\).)44 b(In)33 b(fact)f(w)m(e)i(ha)m(v)m(e)315 4128 y(Mor)491 4143 y Fr(C)536 4128 y Fx(\()p Ft(C)r(;)17 b Fu(G)6 b Fx(\()p Ft(h)p Fx(\)\)\()p Fu(G)g Fx(\()p Ft(g)t Fx(\))p Ft(\023)p Fx(\))27 b(=)g Fu(G)6 b Fx(\()p Ft(h)p Fx(\))p Fu(G)g Fx(\()p Ft(g)t Fx(\))p Ft(\023)28 b Fx(=)g Fu(G)6 b Fx(\()p Ft(hg)t Fx(\))p Ft(\023)27 b Fx(=)h Fu(G)6 b Fx(\(Mor)2604 4143 y Fr(C)2649 4128 y Fx(\()p Fu(F)k Fx(\()p Ft(C)d Fx(\))p Ft(;)17 b(h)p Fx(\)\()p Ft(g)t Fx(\)\))p Ft(\023:)0 4285 y Fx(Hence)37 b(for)e(all)f Ft(C)40 b Fu(2)34 b(C)42 b Fx(the)36 b(functor)g(Mor)1577 4300 y Fr(C)1622 4285 y Fx(\()p Ft(C)r(;)17 b Fu(G)6 b Fx(\(-)o(\)\))33 b(:)g Fu(D)j(\000)-60 b(!)33 b Fv(Set)j Fx(induced)h(b)m(y)f(the)h(bifunctor)0 4401 y(Mor)176 4416 y Fr(C)221 4401 y Fx(\(-)p Ft(;)17 b Fu(G)6 b Fx(\(-)o(\)\))29 b(:)f Fu(C)689 4365 y Fq(op)786 4401 y Fu(\002)23 b(D)32 b(\000)-60 b(!)28 b Fv(Set)34 b Fx(is)e(represen)m(table.)47 b(By)34 b(Theorem)f(8.9.5)g(there)h(is)f(a)g(functor)0 4518 y Fu(F)k Fx(:)28 b Fu(C)34 b(\000)-60 b(!)28 b(D)35 b Fx(suc)m(h)f(that)e(Mor)1114 4533 y Fr(C)1159 4518 y Fx(\(-)p Ft(;)17 b Fu(G)6 b Fx(\(-)o(\)\))1513 4490 y Fu(\030)1513 4522 y Fx(=)1618 4518 y(Mor)1794 4533 y Fr(D)1855 4518 y Fx(\()p Fu(F)k Fx(\(-)n(\))p Ft(;)17 b Fx(-\).)149 4634 y(The)34 b(second)g(statemen)m(t)f(follo)m(ws)e (analogously)-8 b(.)p 3529 4634 4 66 v 3533 4571 59 4 v 3533 4634 V 3591 4634 4 66 v 149 4796 a Fv(Remark)37 b(8.9.17.)50 b Fx(One)45 b(can)g(c)m(haracterize)g(the)g(prop)s(erties) g(that)g Fu(G)54 b Fx(:)49 b Fu(D)i(\000)-60 b(!)48 b(C)j Fx(\(resp.)0 4912 y Fu(F)k Fx(:)46 b Fu(C)52 b(\000)-60 b(!)46 b(D)s Fx(\))d(m)m(ust)g(ha)m(v)m(e)h(in)f(order)g(to)g(p)s (ossess)i(a)e(left-\(righ)m(t-\)adjoin)m(t)c(functor.)76 b(One)43 b(of)0 5028 y(the)33 b(essen)m(tial)g(prop)s(erties)f(for)h (this)f(is)g(that)h Fu(G)39 b Fx(preserv)m(es)c(limits)30 b(\(hence)k(direct)e(pro)s(ducts)i(and)0 5145 y(di\013erence)f(k)m (ernels\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF