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Ff(\001)p Fe(\012)p Ff(1)p Fe(\012)p Ff(1)2437 2107 y Fi(B)k Fh(\012)c Fi(M)33 b Fh(\012)22 b Fi(B)27 b Fh(\012)c Fi(N)p 1915 2079 494 4 v 2326 2077 a Fg(-)2052 2051 y Ff(1)p Fe(\012)p Fd(\034)8 b Fe(\012)p Ff(1)3613 2107 y Fi(M)32 b Fh(\012)23 b Fi(N)p 3182 2079 402 4 v 3501 2077 a Fg(-)3313 2045 y Fd(\026)p Fe(\012)p Fd(\026)p 356 2009 4 1034 v 358 2009 a Fg(?)397 1511 y Fe(r\012)p Ff(1)p Fe(\012)p Ff(1)p 1331 1229 4 254 v 1333 1229 a Fg(?)1372 1121 y Ff(\001)p Fe(\012)p Ff(1)p Fe(\012)p Ff(1)p Fe(\012)p Ff(1)p Fe(\012)p Ff(1)p 2696 1229 V 2698 1229 a Fg(?)2737 1121 y Ff(\001)p Fe(\012)p Ff(1)p Fe(\012)p Ff(1)p Fe(\012)p Ff(1)p Fe(\012)p Ff(1)p 3768 1229 V 3770 1229 a Fg(?)3809 1121 y Ff(\001)p Fe(\012)p Ff(1)p Fe(\012)p Ff(1)p 1331 1619 V 1333 1619 a Fg(?)1372 1510 y Ff(1)p Fe(\012)p Fd(\034)8 b Fe(\012)p Ff(1)p Fe(\012)p Ff(1)p Fe(\012)p Ff(1)p 2696 1619 V 2698 1619 a Fg(?)2737 1509 y Ff(1)p Fe(\012)p Fd(\034)g Ff(\()p Fd(B)s(;B)s Fe(\012)p Fd(M)f Ff(\))p Fe(\012)p Ff(1)p Fe(\012)p Ff(1)p 3768 1619 V 3770 1619 a Fg(?)3809 1510 y Ff(1)p Fe(\012)p Fd(\034)h Fe(\012)p Ff(1)p 1331 2009 V 1333 2009 a Fg(?)1372 1901 y Fe(r\012r\012)p Ff(1)p Fe(\012)p Ff(1)p 2696 2009 V 2698 2009 a Fg(?)2737 1901 y Fe(r\012)p Ff(1)p Fe(\012r\012)p Ff(1)p 3768 2009 V 3770 2009 a Fg(?)3809 1896 y Fd(\026)p Fe(\012)p Fd(\026)0 2251 y Fn(The)34 b(unit)d(la)m(w)i(is)f(the)h(comm)m(utativit)m(y)d(of) 699 2440 y Fi(M)j Fh(\012)22 b Fi(N)1042 2412 y Fh(\030)1042 2444 y Fn(=)1147 2440 y Fc(K)45 b Fh(\012)23 b Fi(M)33 b Fh(\012)23 b Fi(N)636 b(B)27 b Fh(\012)c Fi(M)33 b Fh(\012)22 b Fi(N)p 1690 2411 569 4 v 2176 2409 a Fg(-)1864 2378 y Fd(\021)r Fe(\012)p Ff(1)p Fe(\012)p Ff(1)1116 2830 y Fc(K)45 b Fh(\012)23 b Fc(K)46 b Fh(\012)22 b Fi(M)33 b Fh(\012)23 b Fi(N)367 b(B)28 b Fh(\012)22 b Fi(B)28 b Fh(\012)22 b Fi(M)33 b Fh(\012)23 b Fi(N)p 1858 2801 301 4 v 2076 2799 a Fg(-)1852 2768 y Fd(\021)r Fe(\012)p Fd(\021)r Fe(\012)p Ff(1)p Fe(\012)p Ff(1)697 3220 y Fi(M)33 b Fh(\012)22 b Fi(N)1039 3192 y Fh(\030)1040 3224 y Fn(=)1145 3220 y Fc(K)45 b Fh(\012)23 b Fi(M)33 b Fh(\012)22 b Fc(K)46 b Fh(\012)23 b Fi(N)338 b(B)28 b Fh(\012)22 b Fi(M)33 b Fh(\012)23 b Fi(B)k Fh(\012)c Fi(N)p 1887 3191 272 4 v 2076 3189 a Fg(-)1867 3158 y Fd(\021)r Fe(\012)p Ff(1)p Fe(\012)p Fd(\021)r Fe(\012)p Ff(1)p 1373 2731 4 254 v 1375 2731 a Fg(?)1414 2607 y Fe(\030)1414 2628 y Ff(=)p 1373 3121 V 1375 3121 a Fg(?)1414 3013 y Ff(1)p Fe(\012)p Fd(\034)8 b Fe(\012)p Ff(1)p 2543 2731 V 2545 2731 a Fg(?)2584 2624 y Ff(\001)p Fe(\012)p Ff(1)p Fe(\012)p Ff(1)p 2543 3121 V 2545 3121 a Fg(?)2584 3013 y Ff(1)p Fe(\012)p Fd(\034)g Fe(\012)p Ff(1)p 2543 3511 V 2545 3511 a Fg(?)2584 3398 y Fd(\026)p Fe(\012)p Fd(\026)p 837 3121 4 644 v 839 3121 a Fg(?)745 2812 y Ff(=)2387 3609 y Fi(M)33 b Fh(\012)23 b Fi(N)1663 3406 y Ff(1)1258 3279 y Fg(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 Fn(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 Fc(K)56 b Fn(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 Fk(De\014nition)37 b(8.7.3.)49 b Fn(1.)41 b(Let)27 b(\()p Fi(B)5 b(;)17 b Fh(r)p Fi(;)g(\021)t(;)g Fn(\001)p Fi(;)g(\017)p Fn(\))26 b(b)s(e)g(a)h(bialgebra.)39 b(Let)27 b Fi(A)g Fn(b)s(e)g(a)f(left)g Fi(B)5 b Fn(-mo)s(dule)0 4169 y(with)31 b(structure)i(map)e Fi(\026)c Fn(:)h Fi(B)d Fh(\012)c Fi(A)28 b Fh(\000)-60 b(!)27 b Fi(A)p Fn(.)44 b(Let)31 b(furthermore)h(\()p Fi(A;)17 b Fh(r)2531 4184 y Fd(A)2587 4169 y Fi(;)g(\021)2679 4184 y Fd(A)2736 4169 y Fn(\))32 b(b)s(e)g(an)f(algebra)f(suc)m(h)0 4286 y(that)37 b Fh(r)299 4301 y Fd(A)393 4286 y Fn(and)g Fi(\021)635 4301 y Fd(A)729 4286 y Fn(are)g(homomorphisms)d(of)j Fi(B)5 b Fn(-mo)s(dules.)55 b(Then)38 b(\()p Fi(A;)17 b Fh(r)2785 4301 y Fd(A)2842 4286 y Fi(;)g(\021)2934 4301 y Fd(A)2991 4286 y Fi(;)g(\026)p Fn(\))36 b(is)g(called)g(a)0 4402 y Fi(B)5 b Fj(-mo)-5 b(dule)34 b(algebr)-5 b(a)p Fn(.)149 4518 y(2.)41 b(Let)26 b(\()p Fi(B)5 b(;)17 b Fh(r)p Fi(;)g(\021)t(;)g Fn(\001)p Fi(;)g(\017)p Fn(\))26 b(b)s(e)g(a)f(bialgebra.)40 b(Let)26 b Fi(C)32 b Fn(b)s(e)26 b(a)g(left)f Fi(B)5 b Fn(-mo)s(dule)24 b(with)i(structure)h(map)0 4634 y Fi(\026)g Fn(:)h Fi(B)18 b Fh(\012)12 b Fi(C)35 b Fh(\000)-60 b(!)28 b Fi(C)7 b Fn(.)41 b(Let)28 b(furthermore)g(\()p Fi(C)r(;)17 b Fn(\001)1657 4649 y Fd(C)1715 4634 y Fi(;)g(")1805 4649 y Fd(C)1864 4634 y Fn(\))28 b(b)s(e)g(a)f(coalgebra)g(suc)m(h)i (that)e(\001)3065 4649 y Fd(C)3153 4634 y Fn(and)g Fi(")3383 4649 y Fd(C)3470 4634 y Fn(are)0 4751 y(homomorphisms)j(of)i Fi(B)5 b Fn(-)33 b(mo)s(dules.)42 b(Then)34 b(\()p Fi(C)r(;)17 b Fn(\001)1893 4766 y Fd(C)1951 4751 y Fi(;)g(")2041 4766 y Fd(C)2100 4751 y Fi(;)g(\026)p Fn(\))32 b(is)g(called)f(a)i Fi(B)5 b Fj(-mo)-5 b(dule)34 b(c)-5 b(o)g(algebr)g(a)p Fn(.)149 4867 y(3.)51 b(Let)36 b(\()p Fi(B)5 b(;)17 b Fh(r)p Fi(;)g(\021)t(;)g Fn(\001)p Fi(;)g(\017)p Fn(\))34 b(b)s(e)h(a)g(bialgebra.)49 b(Let)36 b Fi(A)f Fn(b)s(e)g(a)g(left)f Fi(B)5 b Fn(-como)s(dule)34 b(with)h(structure)0 4983 y(map)g Fi(\016)i Fn(:)c Fi(A)g Fh(\000)-60 b(!)33 b Fi(B)c Fh(\012)c Fi(A)p Fn(.)53 b(Let)35 b(furthermore)h(\()p Fi(A;)17 b Fh(r)1932 4998 y Fd(A)1988 4983 y Fi(;)g(\021)2080 4998 y Fd(A)2137 4983 y Fn(\))36 b(b)s(e)g(an)f(algebra)g(suc)m(h)i (that)e Fh(r)3350 4998 y Fd(A)3443 4983 y Fn(and)0 5099 y Fi(\021)48 5114 y Fd(A)139 5099 y Fn(are)g(homomorphisms)d(of)i Fi(B)5 b Fn(-como)s(dules.)48 b(Then)35 b(\()p Fi(A;)17 b Fh(r)2270 5114 y Fd(A)2327 5099 y Fi(;)g(\021)2419 5114 y Fd(A)2476 5099 y Fi(;)g(\016)t Fn(\))34 b(is)g(called)f(a)h Fi(B)5 b Fj(-c)-5 b(omo)g(dule)0 5216 y(algebr)g(a)p Fn(.)p eop %%Page: 36 4 36 3 bop 0 236 a Fl(36)1456 b(8.)77 b(TOOLBO)n(X)149 425 y Fn(4.)51 b(Let)35 b(\()p Fi(B)5 b(;)17 b Fh(r)p Fi(;)g(\021)t(;)g Fn(\001)p Fi(;)g(\017)p Fn(\))34 b(b)s(e)h(a)g (bialgebra.)48 b(Let)35 b Fi(C)42 b Fn(b)s(e)35 b(a)g(left)f Fi(B)5 b Fn(-como)s(dule)33 b(with)i(structure)0 541 y(map)e Fi(\016)g Fn(:)c Fi(C)37 b Fh(\000)-60 b(!)29 b Fi(B)f Fh(\012)23 b Fi(C)7 b Fn(.)47 b(Let)33 b(furthermore)h(\()p Fi(C)r(;)17 b Fn(\001)1907 556 y Fd(C)1965 541 y Fi(;)g(")2055 556 y Fd(C)2114 541 y Fn(\))33 b(b)s(e)h(a)f(coalgebra)g(suc)m(h)i (that)e(\001)3350 556 y Fd(C)3443 541 y Fn(and)0 658 y Fi(")46 673 y Fd(C)139 658 y Fn(are)i(homomorphisms)d(of)i Fi(B)5 b Fn(-como)s(dules.)48 b(Then)36 b(\()p Fi(C)r(;)17 b Fn(\001)2268 673 y Fd(C)2327 658 y Fi(;)g(")2417 673 y Fd(C)2475 658 y Fi(;)g(\016)t Fn(\))34 b(is)g(called)g(a)g Fi(B)5 b Fj(-c)-5 b(omo)g(dule)0 774 y(c)g(o)g(algebr)g(a)p Fn(.)149 948 y Fk(Remark)37 b(8.7.4.)50 b Fn(If)38 b(\()p Fi(C)r(;)17 b Fn(\001)1223 963 y Fd(C)1281 948 y Fi(;)g(")1371 963 y Fd(C)1430 948 y Fn(\))38 b(is)g(a)g Fc(K)17 b Fn(-coalgebra)43 b(and)c(\()p Fi(C)r(;)17 b(\026)p Fn(\))37 b(is)g(a)h Fi(B)5 b Fn(-mo)s(dule,)38 b(then)0 1064 y(\()p Fi(C)r(;)17 b Fn(\001)235 1079 y Fd(C)294 1064 y Fi(;)g(")384 1079 y Fd(C)442 1064 y Fi(;)g(\026)p Fn(\))32 b(is)g(a)g Fi(B)5 b Fn(-mo)s(dule)31 b(coalgebra)h(i\013)f Fi(\026)h Fn(is)g(a)h (homomorphism)d(of)i Fc(K)17 b Fn(-coalgebras.)149 1181 y(If)34 b(\()p Fi(A;)17 b Fh(r)486 1196 y Fd(A)543 1181 y Fi(;)g(\021)635 1196 y Fd(A)692 1181 y Fn(\))34 b(is)f(a)h Fc(K)17 b Fn(-algebra)39 b(and)34 b(\()p Fi(A;)17 b(\016)t Fn(\))33 b(is)h(a)f Fi(B)5 b Fn(-como)s(dule,)33 b(then)i(\()p Fi(A;)17 b Fh(r)3084 1196 y Fd(A)3140 1181 y Fi(;)g(\021)3232 1196 y Fd(A)3289 1181 y Fi(;)g(\016)t Fn(\))34 b(is)f(a)0 1297 y Fi(B)5 b Fn(-como)s(dule)31 b(algebra)g(i\013)h Fi(\016)k Fn(is)c(a)h(homomorphism)d(of)i Fc(K)17 b Fn(-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 Fj(not)f Fn(hold.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF