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b(Let)31 b Fn(B)i Fs(:=)27 b Fm(f)728 1894 y Fc(P)833 1998 y Fl(n;)890 1982 y Fh(\026)896 1998 y Fl(i)940 1969 y Fn(v)987 1984 y Fl(i)1011 1993 y Fe(1)1069 1969 y Fm(\001)17 b Fn(:)g(:)g(:)h Fm(\001)g Fn(v)1340 1984 y Fl(i)1364 1992 y Fg(n)1411 1969 y Fm(j)1431 1946 y Fs(\026)1439 1969 y Fn(i)27 b Fs(=)h(\()p Fn(i)1674 1984 y Fh(1)1713 1969 y Fn(;)17 b(:)g(:)g(:)33 b(;)17 b(i)1982 1984 y Fl(n)2029 1969 y Fs(\))32 b(m)m(ultiindex)f(of)h (length)g Fn(n)p Fm(g)p Fs(.)43 b(Ob)m(viously)0 2094 y Fn(B)d Fs(is)34 b(the)h(subalgebra)f(of)g Fn(T)14 b Fs(\()p Fn(V)21 b Fs(\))35 b(generated)g(b)m(y)h(the)f(elemen)m(ts)g (of)f Fn(V)21 b Fs(.)50 b(Let)34 b Fn(j)k Fs(:)31 b Fn(B)36 b Fm(\000)-60 b(!)31 b Fn(T)14 b Fs(\()p Fn(V)21 b Fs(\))35 b(b)s(e)0 2210 y(the)k(em)m(b)s(edding)f(homomorphism.)60 b(Then)39 b Fn(\023)g Fs(:)f Fn(V)60 b Fm(\000)-60 b(!)38 b Fn(T)14 b Fs(\()p Fn(V)21 b Fs(\))39 b(factors)g(through)f(a)h (linear)e(map)0 2326 y Fn(\023)34 2290 y Fi(0)85 2326 y Fs(:)28 b Fn(V)49 b 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Fn(g)f Fs(:)d Fn(S)6 b Fs(\()p Fn(V)21 b Fs(\))32 b Fm(\000)-60 b(!)32 b Fn(D)s Fs(\()p Fn(V)21 b Fs(\))35 b(suc)m(h)h(that)0 5216 y Fn(g)25 b Fm(\016)d Fn(\023)29 b Fs(=)e Fn(j)6 b Fs(.)43 b(Since)33 b Fn(j)39 b Fs(is)32 b(injectiv)m(e)g(so)h(is)f Fn(\023)p Fs(.)p eop %%Page: 22 8 22 7 bop 0 236 a Fq(22)1456 b(8.)77 b(TOOLBO)n(X)149 425 y Fs(2.)43 b(Let)31 b Fn(B)i Fs(:=)27 b Fm(f)728 350 y Fc(P)833 454 y Fl(n;)890 439 y Fh(\026)896 454 y Fl(i)940 425 y Fn(v)987 440 y Fl(i)1011 449 y Fe(1)1069 425 y Fm(\001)17 b Fn(:)g(:)g(:)h Fm(\001)g Fn(v)1340 440 y Fl(i)1364 448 y Fg(n)1411 425 y Fm(j)1431 403 y Fs(\026)1439 425 y Fn(i)27 b Fs(=)h(\()p Fn(i)1674 440 y Fh(1)1713 425 y Fn(;)17 b(:)g(:)g(:)33 b(;)17 b(i)1982 440 y Fl(n)2029 425 y Fs(\))32 b(m)m(ultiindex)f(of)h(length)g Fn(n)p Fm(g)p Fs(.)43 b(Ob)m(viously)0 550 y Fn(B)d Fs(is)35 b(the)g(subalgebra)g(of)f Fn(S)6 b Fs(\()p Fn(V)22 b Fs(\))34 b(generated)i(b)m(y)g(the)g(elemen)m(ts)f(of)g Fn(V)21 b Fs(.)51 b(Let)35 b Fn(j)j Fs(:)32 b 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y(\000)1821 1376 y(\000)-83 b(\011)0 1618 y Fs(w)m(e)38 b(ha)m(v)m(e)g(id)458 1633 y Fl(B)535 1618 y Fm(\016)p Fn(\023)619 1582 y Fi(0)678 1618 y Fs(=)d Fn(\023)823 1582 y Fi(0)846 1618 y Fs(,)j Fn(p)f Fs(with)f Fn(p)25 b Fm(\016)g Fn(j)31 b Fm(\016)25 b Fn(\023)1552 1582 y Fi(0)1611 1618 y Fs(=)34 b Fn(p)25 b Fm(\016)g Fn(\023)35 b Fs(=)g Fn(\023)2084 1582 y Fi(0)2145 1618 y Fs(exists)i(since)g Fn(\023)2695 1582 y Fi(0)2756 1618 y Fs(is)f(a)h(homomorphism)0 1734 y(of)47 b Fo(K)18 b Fs(-mo)s(dules)52 b(satisfying)47 b Fn(\023)1118 1698 y Fi(0)1141 1734 y Fs(\()p Fn(v)t Fs(\))32 b Fm(\001)g Fn(\023)1394 1698 y Fi(0)1418 1734 y Fs(\()p Fn(v)1507 1698 y Fi(0)1530 1734 y Fs(\))53 b(=)h Fn(\023)1785 1698 y Fi(0)1808 1734 y Fs(\()p Fn(v)1897 1698 y Fi(0)1920 1734 y Fs(\))33 b Fm(\001)f Fn(\023)2085 1698 y Fi(0)2109 1734 y Fs(\()p Fn(v)t Fs(\))47 b(for)g(all)e Fn(v)t(;)17 b(v)2743 1698 y Fi(0)2819 1734 y Fm(2)54 b Fn(V)22 b Fs(.)88 b(Because)49 b(of)0 1850 y Fn(j)6 b(p)23 b Fm(\016)f 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Fs(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 Fc(P)2315 3760 y Fl(n;)2372 3745 y Fh(\026)2378 3760 y Fl(i)2423 3731 y Fn(v)2470 3746 y Fl(i)2494 3755 y Fe(1)2559 3731 y Fm(\001)26 b Fn(:)17 b(:)g(:)25 b Fm(\001)h Fn(v)2854 3746 y Fl(i)2878 3754 y Fg(n)2925 3731 y Fs(.)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 Fn(f)11 b Fs(\()p Fn(v)t Fs(\))j Fm(\001)g Fn(f)d Fs(\()p Fn(v)2287 3820 y Fi(0)2308 3856 y Fs(\))28 b(=)g Fn(f)11 b Fs(\()p Fn(v)2626 3820 y Fi(0)2648 3856 y Fs(\))j Fm(\001)g Fn(f)d Fs(\()p Fn(v)t Fs(\))27 b(for)h(all)e Fn(v)t(;)17 b(v)3377 3820 y Fi(0)3427 3856 y Fm(2)28 b Fn(V)0 3972 y Fs(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 Fp(Prop)s(osition)36 b(8.5.13.)49 b Fk(Given)34 b(a)f Fo(K)18 b Fk(-mo)-5 b(dule)39 b Fn(V)21 b Fk(.)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 Fs(\()p Fn(S)6 b Fs(\()p Fn(V)21 b Fs(\))p Fn(;)c(\023)p Fs(\))p Fk(.)149 4440 y Ff(Pr)n(oof.)42 b Fs(De\014ne)35 b Fn(S)6 b Fs(\()p Fn(V)21 b Fs(\))29 b(:=)h Fn(T)14 b Fs(\()p Fn(V)21 b Fs(\))p Fn(=I)41 b Fs(where)35 b Fn(I)i Fs(=)29 b Fm(h)p Fn(v)t(v)2187 4403 y Fi(0)2233 4440 y Fm(\000)23 b Fn(v)2384 4403 y Fi(0)2407 4440 y Fn(v)t Fm(j)p Fn(v)t(;)17 b(v)2632 4403 y Fi(0)2684 4440 y Fm(2)29 b Fn(V)22 b Fm(i)33 b Fs(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 Fn(v)t(v)1472 4520 y Fi(0)1513 4556 y Fm(\000)19 b Fn(v)1660 4520 y Fi(0)1683 4556 y Fn(v)t Fs(.)42 b(Let)31 b Fn(\023)g Fs(b)s(e)g(the)g(canonical)e(map)h Fn(V)49 b Fm(\000)-60 b(!)28 b Fn(T)14 b Fs(\()p Fn(V)21 b Fs(\))28 b Fm(\000)-60 b(!)0 4672 y Fn(S)6 b Fs(\()p Fn(V)21 b Fs(\).)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 Fp(Prop)s(osition)k(8.5.14.)49 b Fk(The)54 b(c)-5 b(onstruction)54 b(of)g(symmetric)g(algebr)-5 b(as)53 b Fn(S)6 b Fs(\()p Fn(V)21 b Fs(\))54 b Fk(de\014nes)f(a)0 5080 y(functor)45 b Fn(S)52 b Fs(:)47 b Fo(K)18 b Fk(-)5 b Fp(Mo)s(d)48 b Fm(\000)-58 b(!)47 b Fo(K)17 b Fk(-)6 b Fp(cAlg)44 b Fk(that)h(is)g(left)g(adjoint)g(to)g(the)g(underlying)g (functor)g Fn(U)57 b Fs(:)0 5196 y Fo(K)17 b Fk(-)6 b Fp(cAlg)27 b Fm(\000)-58 b(!)28 b Fo(K)17 b Fk(-)6 b Fp(Mo)s(d)q Fk(.)p eop %%Page: 23 9 23 8 bop 1530 236 a Fq(5.)77 b(ALGEBRAS)1427 b(23)149 425 y Ff(Pr)n(oof.)42 b Fs(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 Fp(Problem)k(8.5.3.)49 b Fs(Let)42 b Fn(X)49 b Fs(b)s(e)41 b(a)g(set)h(and)f Fn(V)64 b Fs(:=)42 b Fo(K)18 b Fn(X)55 b Fs(b)s(e)41 b(the)h(free)f Fo(K)18 b Fs(-mo)s(dule)45 b(o)m(v)m(er)d Fn(X)8 b Fs(.)0 718 y(Sho)m(w)32 b(that)f Fn(X)k Fm(\000)-60 b(!)28 b Fn(V)49 b Fm(\000)-60 b(!)28 b Fn(S)6 b Fs(\()p Fn(V)21 b Fs(\))31 b(de\014nes)h(a)f Fk(fr)-5 b(e)g(e)33 b(c)-5 b(ommutative)33 b(algebr)-5 b(a)30 b Fs(o)m(v)m(er)i Fn(X)8 b Fs(,)31 b(i.e.)43 b(for)30 b(ev)m(ery)0 835 y(comm)m(utativ)m(e)21 b Fo(K)d Fs(-algebra)26 b Fn(A)d Fs(and)f(ev)m(ery)i(map)e Fn(f)38 b Fs(:)28 b Fn(X)35 b Fm(\000)-60 b(!)28 b Fn(A)22 b Fs(there)h(is)f(a)g(unique)h (homomorphism)0 951 y(of)32 b Fo(K)18 b Fs(-algebras)37 b Fn(g)31 b Fs(:)d Fn(S)6 b Fs(\()p Fn(V)21 b Fs(\))28 b Fm(\000)-60 b(!)27 b Fn(A)33 b Fs(suc)m(h)h(that)e(the)h(diagram)1479 1104 y Fn(X)341 b(S)6 b Fs(\()p Fn(V)21 b Fs(\))p 1597 1081 276 4 v 1790 1079 a Fj(-)1648 1348 y Fn(f)1592 1230 y Fj(@)1675 1313 y(@)1758 1396 y(@)1841 1479 y(@)1860 1498 y(@)-83 b(R)1975 1601 y Fn(A)p 2009 1498 4 351 v 2011 1498 a Fj(?)2050 1335 y Fn(g)0 1728 y Fs(comm)m(utes.)149 1844 y(The)40 b(algebra)e Fo(K)18 b Fs([)p Fn(X)8 b Fs(])45 b(:=)38 b Fn(S)6 b Fs(\()p Fo(K)18 b Fn(X)8 b Fs(\))44 b(is)39 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Fs(\).)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 Fn(")f Fs(:)h Fn(S)6 b Fs(\()p Fn(V)21 b Fs(\))38 b Fm(\000)-60 b(!)37 b Fo(K)62 b Fs(with)0 2658 y Fn(")p Fs(\()p Fn(v)t Fs(\))27 b(=)g(0)33 b(for)f(all)e Fn(v)i Fm(2)c Fn(V)21 b Fs(.)149 2774 y(5.)44 b(Sho)m(w)33 b(that)f(\()p Fn(")22 b Fm(\012)h Fs(1\)\001)k(=)h(\(1)22 b Fm(\012)g Fn(")p Fs(\)\001)28 b(=)g(id)1828 2790 y Fl(S)t Fh(\()p Fl(V)16 b Fh(\))1990 2774 y Fs(.)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 Fn(S)34 b Fs(:)27 b Fn(S)6 b Fs(\()p Fn(V)22 b Fs(\))27 b Fm(\000)-60 b(!)28 b Fn(S)6 b Fs(\()p Fn(V)21 b Fs(\))28 b(with)0 3007 y Fn(S)6 b Fs(\()p Fn(v)t Fs(\))27 b(=)h Fm(\000)p Fn(v)t Fs(.)149 3123 y(7.)44 b(Sho)m(w)33 b(that)f(the)h (diagrams)1300 3281 y Fn(S)6 b Fs(\()p Fn(V)21 b Fs(\))241 b Fo(K)p 1548 3258 185 4 v 1650 3256 a Fj(-)1624 3237 y Fl(")2080 3281 y Fn(S)6 b Fs(\()p Fn(V)21 b Fs(\))p 1867 3258 V 1969 3256 a Fj(-)1940 3224 y Fl(\021)1129 3671 y Fn(S)6 b Fs(\()p Fn(V)21 b Fs(\))h Fm(\012)h Fn(S)6 b Fs(\()p Fn(V)21 b Fs(\))218 b Fn(S)6 b Fs(\()p Fn(V)21 b Fs(\))h Fm(\012)h Fn(S)6 b Fs(\()p Fn(V)21 b Fs(\))p 1720 3648 160 4 v 1797 3646 a Fj(-)1732 3620 y Fh(1)p Fi(\012)p Fl(S)1732 3710 y(S)t Fi(\012)p Fh(1)p 1408 3578 4 254 v 1410 3578 a Fj(?)1312 3474 y Fh(\001)p 2188 3578 V 2190 3407 a Fj(6)2229 3474 y Fi(r)0 3831 y Fs(comm)m(ute.)149 4008 y Fp(De\014nition)37 b(8.5.15.)49 b Fs(Let)44 b Fo(K)68 b Fs(b)s(e)45 b(a)f(comm)m(utativ)m(e)f(ring.)78 b(Let)44 b Fn(V)66 b Fs(b)s(e)45 b(a)f Fo(K)17 b Fs(-mo)s(dule.)83 b(A)0 4124 y Fo(K)17 b Fs(-algebra)41 b Fn(E)6 b Fs(\()p Fn(V)21 b Fs(\))36 b(together)h(with)e(a)h(homomorphism)d(of)j Fo(K)17 b Fs(-mo)s(dules)41 b Fn(\023)34 b Fs(:)f Fn(V)56 b Fm(\000)-60 b(!)33 b Fn(E)6 b Fs(\()p Fn(V)22 b Fs(\),)36 b(suc)m(h)0 4241 y(that)e Fn(\023)p Fs(\()p Fn(v)t Fs(\))374 4204 y Fh(2)443 4241 y Fs(=)c(0)j(for)h(all)d Fn(v)j Fm(2)c Fn(V)22 b Fs(,)34 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Fs(-mo)s(dule)d(in)i(the)h(sense)i(of)d(De\014nition)e(8.5.19.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF