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b(sub)q(ob)s(ject)h(will)g(b)q(e)g(used)18 850 y(in)f(the)g(sense)g(of) f([2],)g(that)g(is)h(as)f(a)h(represen)o(tativ)o(e)f(of)g(the)h(usual)g (equiv)m(alence)i(class)e(of)18 904 y(monomorphisms.)76 958 y(In)16 b(the)f(follo)o(wing)i(de\014nition)g Fn(p)629 965 y Fk(A)669 958 y Fu(:)12 b Fn(A)f Fo(\002)f Fn(B)15 b Fo(\000)-7 b(!)12 b Fn(A)k Fu(\(resp.)k Fn(p)1125 965 y Fk(B)1166 958 y Fu(:)13 b Fn(A)d Fo(\002)h Fn(B)k Fo(\000)-8 b(!)13 b Fn(B)r Fu(\))i(denotes)18 1012 y(the)g(canonical)i(pro)s (jection.)18 1120 y Fv(De\014nition)i(2.1)55 b Fu(1.)18 b(A)c(sub)q(ob)s(ject)h Fn(\023)709 1127 y Fk(X)753 1120 y Fu(:)d Fn(X)k(,)-8 b Fo(!)12 b Fn(A)d Fo(\002)f Fn(B)16 b Fu(is)f(called)h(a)e Fe(we)n(ak)i(c)n(omplement)f(of)109 1174 y Fn(B)r Fu(,)f(if)i(the)f(morphism)h Fn(X)577 1148 y Fk(\023)589 1152 y Fj(X)560 1174 y Fo(\000)-8 b(!)12 b Fn(A)f Fo(\002)f Fn(B)801 1147 y Fk(p)818 1151 y Fj(A)784 1174 y Fo(\000)-7 b(!)12 b Fn(A)k Fu(has)f(k)o(ernel)h(0)c Fo(\000)-7 b(!)12 b Fn(X)t Fu(.)55 1269 y(2.)18 b(A)f(sub)q(ob)s(ject)g Fn(\023)385 1276 y Fk(X)432 1269 y Fu(:)d Fn(X)19 b(,)-8 b Fo(!)15 b Fn(A)c Fo(\002)h Fn(B)18 b Fu(is)g(called)g(a)e Fe(c)n(omplement)i(of)f Fn(B)r Fu(,)f(if)i(the)e(morphism)109 1323 y Fn(X)180 1297 y Fk(\023)192 1301 y Fj(X)163 1323 y Fo(\000)-8 b(!)12 b Fn(A)f Fo(\002)f Fn(B)404 1296 y Fk(p)421 1300 y Fj(A)387 1323 y Fo(\000)-7 b(!)12 b Fn(A)k Fu(is)f(an)g(isomorphism.)55 1418 y(3.)j Fn(U)i Fu(is)c(called)g Fe(internal)g(dir)n(e)n(ct)g(pr)n(o)n(duct)f Fu(of)g(the)g(sub)q(ob)s(jects)h Fn(A)c(,)-8 b Fo(!)13 b Fn(U)20 b Fu(and)15 b Fn(B)f(,)-8 b Fo(!)13 b Fn(U)20 b Fu(if)150 1514 y(a\))e Fn(A)12 b(,)-8 b Fo(!)13 b Fn(U)20 b Fu(and)15 b Fn(B)g(,)-8 b Fo(!)12 b Fn(U)20 b Fu(are)15 b(k)o(ernels,)148 1586 y(b\))j(the)d(in)o(tersection)h(of)f Fn(A)g Fu(and)g Fn(B)i Fu(is)f Fn(A)10 b Fo(\\)h Fn(B)j Fu(=)f(0,)153 1659 y(c\))18 b(the)j(canonical)h(morphism)f Fn(A)758 1627 y Fm(`)804 1659 y Fn(B)j Fo(\000)-7 b(!)21 b Fn(U)26 b Fu(is)21 b(a)g(di\013erence)h(cok)o(ernel)f(\(a)f(fact)209 1713 y(whic)o(h)c(w)o(e)f(abbreviate)g(b)o(y)g Fn(A)c Fo([)f Fn(B)k Fu(=)f Fn(U)5 b Fu(\).)76 1821 y(Let)16 b Fn(A)h Fu(and)f Fn(B)i Fu(b)q(e)f(ob)s(jects)e(in)i Fo(C)s Fu(.)23 b(W)l(e)16 b(consider)h Fn(B)h Fu(as)e(a)g(sub)q(ob)s (ject)g(of)g Fn(A)11 b Fo(\002)g Fn(B)18 b Fu(via)f(the)18 1875 y(canonical)f(morphism)429 1873 y Fm(e)430 1875 y Fn(\023)446 1882 y Fk(B)487 1875 y Fu(:)c Fn(B)j Fo(\000)-7 b(!)12 b Fn(A)e Fo(\002)g Fn(B)17 b Fu(induced)g(b)o(y)e(id)f(:)e Fn(B)j Fo(\000)-8 b(!)12 b Fn(B)17 b Fu(and)f(0)c(:)g Fn(B)j Fo(\000)-7 b(!)12 b Fn(A)p Fu(.)18 2001 y Fv(Lemma)17 b(2.2)h Fn(B)352 1975 y Fm(e)-19 b Fk(\023)368 1979 y Fj(B)338 2001 y Fo(\000)-8 b(!)12 b Fn(A)f Fo(\002)f Fn(B)579 1974 y Fk(p)596 1978 y Fj(A)562 2001 y Fo(\000)-7 b(!)12 b Fn(A)17 b Fe(is)e(a)i(kernel)e(diagr)n(am.)76 2109 y Fd(Pro)q(of:)39 b Fu(If)15 b Fn(g)f Fu(:)e Fn(X)k Fo(\000)-8 b(!)13 b Fn(A)c Fo(\002)g Fn(B)16 b Fu(is)f(giv)o(en)g(with) g Fn(p)911 2116 y Fk(A)938 2109 y Fn(g)f Fu(=)f(0)i(then)f Fn(p)1185 2116 y Fk(A)1212 2109 y Fn(g)g Fu(=)f Fn(p)1319 2116 y Fk(A)1345 2107 y Fm(e)1346 2109 y Fn(\023)1362 2116 y Fk(B)1391 2109 y Fn(p)1414 2116 y Fk(B)1442 2109 y Fn(g)h Fu(=)f(0)i(and)18 2163 y Fn(p)41 2170 y Fk(B)69 2163 y Fn(g)h Fu(=)f Fn(p)180 2170 y Fk(B)207 2161 y Fm(e)208 2163 y Fn(\023)224 2170 y Fk(B)253 2163 y Fn(p)276 2170 y Fk(B)305 2163 y Fn(g)i Fu(implies)i Fn(g)c Fu(=)586 2161 y Fm(e)587 2163 y Fn(\023)603 2170 y Fk(B)632 2163 y Fn(p)655 2170 y Fk(B)683 2163 y Fn(g)r Fu(,)h(a)g(factorization)g(of) g Fn(g)i Fu(through)1305 2161 y Fm(e)1306 2163 y Fn(\023)1322 2170 y Fk(B)1350 2163 y Fu(.)23 b(Since)1505 2161 y Fm(e)1506 2163 y Fn(\023)1522 2170 y Fk(B)1567 2163 y Fu(is)17 b(a)18 2217 y(section)f(this)f(factorization)g(is)h(unique.)p 1613 2190 24 2 v 1613 2215 2 25 v 1635 2215 V 1613 2217 24 2 v 18 2370 a Fv(Prop)q(osition)i(2.3)g Fn(\023)401 2377 y Fk(X)455 2370 y Fu(:)j Fn(X)k(,)-8 b Fo(!)21 b Fn(A)14 b Fo(\002)g Fn(B)23 b Fe(is)e(a)g(we)n(ak)h(c)n(omplement)e(of) i Fn(B)h Fe(if)e(and)g(only)g(if)18 2424 y Fn(X)13 b Fo(\\)e Fn(B)j Fu(=)f(0)p Fe(.)611 2738 y Ff(krs.tex)23 b(-)g(Date:)47 b(June)24 b(23,)f(1994)g(Time:)47 b(10:45)p eop %%Page: 4 4 4 3 bop 18 54 a Fu(4)533 b Fh(Bo)q(do)11 b(P)o(areigis)f(and)g(Helm)o (ut)h(R\177)-18 b(ohrl)76 145 y Fd(Pro)q(of:)39 b Fu(Let)16 b Fn(X)i Fu(b)q(e)e(a)f(w)o(eak)g(complemen)o(t)g(of)g Fn(B)r Fu(.)20 b(In)15 b(the)h(comm)o(utativ)o(e)e(diagram)644 228 y Fn(X)f Fo(\\)d Fn(B)147 b(B)p 786 214 118 2 v 862 213 a Fl(-)822 200 y Fk(u)842 204 y Fj(B)515 410 y Fu(0)i Fn(X)p 551 395 123 2 v 632 394 a Fl(-)872 408 y Fn(A)10 b Fo(\002)g Fn(B)p 742 395 117 2 v 817 394 a Fl(-)781 381 y Fk(\023)793 385 y Fj(X)1093 405 y Fn(A;)p 1012 395 69 2 v 1039 394 a Fl(-)1025 380 y Fk(p)1042 384 y Fj(A)570 310 y Fk(\027)635 286 y Fl(\000)593 327 y(\000)558 363 y(\000)-42 b(\011)p 707 363 2 119 v 108 w(?)726 308 y Fk(u)746 312 y Fj(X)p 934 363 V 935 363 a Fl(?)950 316 y Fm(e)-20 b Fk(\023)965 320 y Fj(B)1054 313 y Fi(0)967 286 y Fl(@)1008 327 y(@)1043 363 y(@)-42 b(R)18 477 y Fu(with)17 b Fn(u)149 484 y Fk(B)194 477 y Fu(and)g Fn(u)310 484 y Fk(X)359 477 y Fu(the)g(canonical)h(morphisms,)f(w)o(e)g(ha)o(v)o (e)f Fn(p)1083 484 y Fk(A)1110 477 y Fn(\023)1126 484 y Fk(X)1158 477 y Fn(u)1184 484 y Fk(X)1231 477 y Fu(=)f(0)i(hence)h (there)f(is)g(a)18 531 y(factorization)f Fn(\027)k Fu(of)c Fn(u)404 538 y Fk(X)453 531 y Fu(through)g(0)g(b)o(y)h(the)g(prop)q (ert)o(y)f(of)g(w)o(eak)g(complemen)o(ts.)25 b(Since)18 b Fn(u)1605 538 y Fk(X)18 585 y Fu(is)e(a)e(zero-morphism)i(and)f(a)g (monomorphism)g(w)o(e)g(get)g Fn(X)e Fo(\\)e Fn(B)j Fu(=)f(0.)76 639 y(Let)i Fn(X)f Fo(\\)c Fn(B)15 b Fu(=)d(0.)20 b(In)c(the)f(comm)o (utativ)o(e)g(diagram)746 901 y(0)149 b Fn(X)p 782 886 123 2 v 863 885 a Fl(-)739 1081 y Fn(B)102 b(A)10 b Fo(\002)g Fn(B)p 789 1068 74 2 v 821 1067 a Fl(-)803 1054 y Fm(e)-19 b Fk(\023)819 1058 y Fj(B)1104 1081 y Fn(A)p 1016 1068 75 2 v 1049 1067 a Fl(-)1033 1053 y Fk(p)1050 1057 y Fj(A)517 715 y Fu(Ke\()p Fn(f)5 b Fu(\))786 798 y Fk(g)639 756 y Fl(H)681 777 y(H)722 798 y(H)764 819 y(H)805 839 y(H)834 854 y(H)-42 b(j)607 777 y(@)649 819 y(@)684 854 y(@)g(R)619 896 y Fk(h)592 777 y Fl(A)612 819 y(A)633 860 y(A)654 902 y(A)675 943 y(A)695 985 y(A)716 1026 y(A)721 1035 y(A)-21 b(U)p 757 1035 2 119 v 15 w(?)p 938 1035 V 140 w(?)882 981 y Fk(\023)894 985 y Fj(X)1058 984 y Fk(f)971 959 y Fl(@)1012 1000 y(@)1047 1035 y(@)-42 b(R)18 1177 y Fn(B)82 1152 y Fm(e)-20 b Fk(\023)97 1156 y Fj(B)67 1177 y Fo(\000)-7 b(!)12 b Fn(A)s Fo(\002)s Fn(B)295 1150 y Fk(p)312 1154 y Fj(A)278 1177 y Fo(\000)-7 b(!)12 b Fn(A)g Fu(is)g(a)g(k)o(ernel)g(diagram)g(b)o(y)f(Lemma)h(2.2.) 18 b(Hence)12 b Fn(h)g Fu(can)g(b)q(e)h(constructed)18 1231 y(uniquely)18 b(from)c Fn(g)j Fu(suc)o(h)f(that)550 1229 y Fm(e)551 1231 y Fn(\023)567 1238 y Fk(B)595 1231 y Fn(h)e Fu(=)f Fn(\023)699 1238 y Fk(X)731 1231 y Fn(g)r Fu(.)20 b(No)o(w)15 b Fn(g)i Fu(can)f(b)q(e)g(factored)f(through)g(0)g (and)h(th)o(us)18 1285 y(m)o(ust)e(b)q(e)i(zero.)k(This)c(means)f(that) g Fn(X)j Fu(is)e(a)e(w)o(eak)h(complemen)o(t)h(of)f Fn(B)r Fu(.)p 1613 1259 24 2 v 1613 1283 2 25 v 1635 1283 V 1613 1285 24 2 v 18 1439 a Fv(Prop)q(osition)j(2.4)g Fe(L)n(et)12 b Fn(\023)476 1446 y Fk(X)520 1439 y Fu(:)g Fn(X)k(,)-8 b Fo(!)12 b Fn(A)q Fo(\002)q Fn(B)j Fe(b)n(e)c(a)h(sub)n (obje)n(ct.)19 b(The)12 b(fol)r(lowing)g(ar)n(e)g(e)n(quivalent:)53 1516 y(1.)19 b Fn(X)g Fe(is)d(a)h(c)n(omplement)e(of)i Fn(B)r Fe(.)53 1602 y(2.)i(Ther)n(e)e(is)h(a)g(unique)g(morphism)h Fn(f)i Fu(:)15 b Fn(A)h Fo(\000)-7 b(!)15 b Fn(B)20 b Fe(and)e(an)g(epimorphism)h Fn(g)e Fu(:)f Fn(A)f Fo(\000)-7 b(!)16 b Fn(X)109 1656 y Fe(such)g(that)h(the)g(diagr)n(am)628 1954 y Fn(A)147 b(A)10 b Fo(\002)h Fn(B)p 675 1941 120 2 v 675 1940 a Fl(\033)715 1962 y Fk(p)732 1966 y Fj(A)1081 1954 y Fn(B)p 949 1941 119 2 v 1026 1940 a Fl(-)987 1962 y Fk(p)1004 1966 y Fj(B)855 1729 y Fn(A)703 1837 y Fi(id)799 1787 y Fl(\000)757 1828 y(\000)716 1870 y(\000)677 1909 y(\000)-42 b(\011)1014 1834 y Fk(f)904 1787 y Fl(@)945 1828 y(@)987 1870 y(@)1026 1909 y(@)g(R)851 1842 y Fn(X)p 871 1795 2 50 v 872 1795 a Fl(?)890 1774 y Fk(g)p 871 1909 V 872 1909 a Fl(?)890 1888 y Fk(\023)902 1892 y Fj(X)109 2028 y Fe(c)n(ommutes.)53 2114 y(3.)19 b(The)d(induc)n(e)n(d)f (morphism)i Fn(h)g Fe(in)f(the)g(c)n(ommutative)h(diagr)n(am)634 2210 y Fn(X)143 b(X)13 b Fo(\002)d Fn(B)p 688 2197 114 2 v 688 2196 a Fl(\033)723 2182 y Fk(p)740 2186 y Fj(X)1090 2210 y Fn(B)p 961 2197 116 2 v 1035 2196 a Fl(-)998 2179 y Fk(p)1015 2167 y Fc(0)1015 2188 y Fj(B)637 2437 y Fn(A)147 b(A)10 b Fo(\002)h Fn(B)p 685 2425 120 2 v 685 2424 a Fl(\033)724 2409 y Fk(p)741 2413 y Fj(A)1090 2437 y Fn(B)p 958 2425 119 2 v 1035 2424 a Fl(-)996 2409 y Fk(p)1013 2413 y Fj(B)591 2324 y Fn(A)f Fo(\002)h Fn(B)p 653 2278 2 50 v 654 2278 a Fl(?)672 2257 y Fk(\023)684 2261 y Fj(X)p 653 2392 V 654 2392 a Fl(?)672 2370 y Fk(p)689 2374 y Fj(A)p 881 2392 2 164 v 881 2392 a Fl(?)900 2320 y Fk(h)p 1108 2392 V 1109 2392 a Fl(?)1127 2320 y Fi(id)611 2738 y Ff(krs.tex)23 b(-)g(Date:)47 b(June)24 b(23,)f(1994)g(Time:)47 b(10:45)p eop %%Page: 5 5 5 4 bop 644 54 a Fh(Krull-Sc)o(hmid)o(t)9 b(Theorem)601 b Fu(5)109 145 y Fe(is)16 b(an)g(isomorphism.)76 231 y Fd(Pro)q(of:)35 b Fu(\(1\))9 b Fo(,)i Fu(\(2\):)16 b Fn(p)470 238 y Fk(A)497 231 y Fn(\023)513 238 y Fk(X)555 231 y Fu(is)c(an)e(isomorphism)h(if)h(and)e(only)i(if)f(there)g(is)g (an)f(epimorphism)18 285 y Fn(g)16 b Fu(with)g(\()p Fn(p)201 292 y Fk(A)227 285 y Fn(\023)243 292 y Fk(X)275 285 y Fu(\))p Fn(g)e Fu(=)f(id)q(.)19 b(Th)o(us)d Fn(f)k Fu(can)15 b(b)q(e)h(constructed)f(uniquely)j(suc)o(h)d(that)g Fn(f)i Fu(=)c Fn(p)1491 292 y Fk(B)1520 285 y Fn(\023)1536 292 y Fk(X)1568 285 y Fn(g)r Fu(.)76 339 y(\(1\))h Fo(\))h Fu(\(3\):)k(If)c Fn(p)369 346 y Fk(A)396 339 y Fn(\023)412 346 y Fk(X)459 339 y Fu(is)h(an)f(isomorphism)g(then,)h(ob)o(viously)l (,)f Fn(h)h Fu(is)g(an)f(isomorphism.)76 393 y(\(3\))f Fo(\))h Fu(\(1\):)k(Let)c Fn(h)g Fu(b)q(e)h(an)f(isomorphism.)21 b(In)16 b(the)f(comm)o(utativ)o(e)g(diagram)737 495 y Fn(A)147 b(A)10 b Fo(\002)g Fn(B)p 784 482 120 2 v 862 481 a Fl(-)823 469 y Fm(e)-20 b Fk(\023)838 473 y Fj(A)733 609 y Fn(X)143 b(X)13 b Fo(\002)e Fn(B)p 788 596 114 2 v 788 595 a Fl(\033)822 581 y Fk(p)839 585 y Fj(X)737 836 y Fn(A)147 b(A)10 b Fo(\002)g Fn(B)p 784 823 120 2 v 784 822 a Fl(\033)824 808 y Fk(p)841 812 y Fj(A)p 753 563 2 50 v 754 563 a Fl(?)598 551 y Fk(p)615 555 y Fj(X)642 551 y Fk(h)661 538 y Fc(\000)p Fb(1)696 551 y Fm(e)-19 b Fk(\023)712 555 y Fj(A)691 722 y Fn(A)10 b Fo(\002)g Fn(B)p 753 677 V 754 677 a Fl(?)696 656 y Fk(\023)708 660 y Fj(X)p 753 791 V 754 791 a Fl(?)695 769 y Fk(p)712 773 y Fj(A)p 980 563 V 981 563 a Fl(?)999 552 y Fk(h)1018 539 y Fc(\000)p Fb(1)p 980 791 2 164 v 981 791 a Fl(?)999 719 y Fk(h)18 907 y Fu(w)o(e)18 b(ha)o(v)o(e)f Fn(p)218 914 y Fk(A)244 905 y Fm(e)245 907 y Fn(\023)261 914 y Fk(A)306 907 y Fu(=)h(id)q(.)28 b(Hence)20 b(\()p Fn(A)d Fo(\000)-7 b(!)17 b Fn(X)j Fo(\000)-7 b(!)17 b Fn(A)p Fu(\))g(=)h(id)h(in)g(the)g(ab)q(o)o(v)o(e)e(diagram.) 29 b(Th)o(us)18 961 y(it)17 b(su\016ces)f(to)g(sho)o(w)g(that)g Fn(p)514 968 y Fk(A)541 961 y Fn(\023)557 968 y Fk(X)605 961 y Fu(is)h(a)f(monomorphism.)24 b(Supp)q(ose)18 b(that)e Fn(p)1337 968 y Fk(A)1363 961 y Fn(\023)1379 968 y Fk(X)1411 961 y Fn(f)k Fu(=)15 b Fn(p)1526 968 y Fk(A)1553 961 y Fn(\023)1569 968 y Fk(X)1601 961 y Fn(g)r Fu(.)18 1015 y(Then)h(w)o(e)e(ha)o(v)o(e)h(the)h(comm)o(utativ)o(e)e(diagram)787 1156 y Fl(\000)745 1197 y(\000)704 1239 y(\000)665 1278 y(\000)-42 b(\011)773 1138 y(\000)732 1179 y(\000)690 1221 y(\000)651 1260 y(\000)g(\011)775 1199 y Fk(g)708 1162 y(f)835 1089 y Fn(Y)p 843 1269 V 844 1269 a Fl(?)810 1198 y Fm(e)806 1200 y Fk(f)p 861 1269 V 862 1269 a Fl(?)878 1199 y Fm(e)-21 b Fk(g)995 1196 y Fi(0)885 1147 y Fl(@)927 1188 y(@)968 1230 y(@)1007 1269 y(@)-42 b(R)605 1314 y Fn(X)143 b(X)14 b Fo(\002)c Fn(B)p 660 1301 114 2 v 660 1300 a Fl(\033)695 1322 y Fk(p)712 1326 y Fj(X)1062 1314 y Fn(B)p 933 1301 116 2 v 1007 1300 a Fl(-)970 1333 y Fk(p)987 1321 y Fc(0)987 1342 y Fj(B)609 1541 y Fn(A)147 b(A)10 b Fo(\002)h Fn(B)p 656 1529 120 2 v 656 1528 a Fl(\033)696 1513 y Fk(p)713 1517 y Fj(A)1062 1541 y Fn(B)p 930 1529 119 2 v 1007 1528 a Fl(-)968 1513 y Fk(p)985 1517 y Fj(B)p 625 1496 2 164 v 626 1496 a Fl(?)528 1418 y Fk(p)545 1422 y Fj(A)569 1418 y Fk(\023)581 1422 y Fj(X)p 852 1496 V 853 1496 a Fl(?)871 1424 y Fk(h)p 1080 1496 V 1080 1496 a Fl(?)1099 1424 y Fi(id)18 1623 y Fu(De\014ne)165 1609 y Fm(e)155 1623 y Fn(f)17 b Fu(and)280 1621 y Fm(e)278 1623 y Fn(g)c Fu(b)o(y)f(the)f(univ)o(ersal)i(prop)q(ert)o(y)e(of)g (the)h(pro)q(duct)g Fn(X)6 b Fo(\002)s Fn(B)r Fu(.)18 b(Then)13 b Fn(p)1397 1630 y Fk(A)1423 1623 y Fn(h)1459 1609 y Fm(e)1449 1623 y Fn(f)18 b Fu(=)13 b Fn(p)1560 1630 y Fk(A)1587 1623 y Fn(h)1614 1621 y Fm(e)1613 1623 y Fn(g)18 1677 y Fu(and)k Fn(p)131 1684 y Fk(B)159 1677 y Fn(h)195 1663 y Fm(e)185 1677 y Fn(f)j Fu(=)15 b(0)g(=)g Fn(p)388 1684 y Fk(B)417 1677 y Fn(h)444 1675 y Fm(e)443 1677 y Fn(g)j Fu(and)f(hence)g Fn(h)736 1663 y Fm(e)726 1677 y Fn(f)j Fu(=)c Fn(h)846 1675 y Fm(e)845 1677 y Fn(g)i Fu(and)984 1663 y Fm(e)974 1677 y Fn(f)i Fu(=)1068 1675 y Fm(e)1067 1677 y Fn(g)e Fu(since)f Fn(h)g Fu(is)g(an)g (isomorphism.)18 1731 y(So)e Fn(f)j Fu(=)13 b Fn(g)j Fu(and)g Fn(p)319 1738 y Fk(A)345 1731 y Fn(\023)361 1738 y Fk(X)408 1731 y Fu(is)g(a)f(monomorphism.)p 1613 1704 24 2 v 1613 1729 2 25 v 1635 1729 V 1613 1731 24 2 v 18 1885 a Fv(Remark)i(2.5)g Fe(1\))k(In)e(some)h(sense)f(2.4,)j (\(3\),)e(me)n(ans)g(that)g Fn(A)14 b Fo(\002)f Fn(B)22 b Fe(is)e(gener)n(ate)n(d)f(by)i(the)18 1939 y(sub)n(obje)n(cts)c Fn(X)22 b Fe(and)d Fn(B)r Fe(.)28 b(Observe,)18 b(however,)i(that)f (ther)n(e)g(is)f(also)g(a)h(c)n(anonic)n(al)e(morphism)18 1993 y Fn(X)67 1960 y Fm(`)113 1993 y Fn(B)e Fo(\000)-7 b(!)12 b Fn(A)s Fo(\002)s Fn(B)j Fe(which,)f(in)f(gener)n(al,)f(do)n (es)h(not)g(factor)h(thr)n(ough)g Fn(X)7 b Fo(\002)s Fn(B)14 b Fe(in)f(the)g(c)n(anonic)n(al)18 2046 y(way)k(\(e.g.)j Fa(Gr)o Fe(\))c(nor)g(is)g(it)g(an)g(epimorphism)i(\(e.g.)i(c)n (ommutative)d(monoids\).)76 2100 y(2\))f(In)g(2.4,)h(\(2\),)f(one)h(c)n (an)f(c)n(onsider)g Fn(X)k Fe(as)c(the)h(gr)n(aph)g(of)g(the)g (morphism)g Fn(f)i Fu(:)13 b Fn(A)g Fo(\000)-7 b(!)13 b Fn(B)r Fe(.)18 2154 y(So)j(this)g(p)n(art)h(of)f(Pr)n(op)n(osition)g (2.4)g(may)h(b)n(e)f(r)n(ephr)n(ase)n(d)g(as:)76 2208 y(ther)n(e)f(is)g(a)h(bije)n(ction)e(b)n(etwe)n(en)g(the)i(c)n (omplements)e(of)i Fn(B)h Fe(in)e Fn(A)8 b Fo(\002)g Fn(B)18 b Fe(and)d(the)g(morphisms)18 2262 y Fn(f)j Fu(:)12 b Fn(A)h Fo(\000)-8 b(!)12 b Fn(B)r Fe(.)76 2316 y(T)m(o)i(show)h(that) h(e)n(ach)e Fn(f)20 b Fe(determines)15 b(a)g(sub)n(obje)n(ct)f Fn(X)k Fe(of)d Fn(A)7 b Fo(\002)g Fn(B)r Fe(,)15 b(let)1274 2303 y Fm(e)1265 2316 y Fn(f)i Fu(:)c Fn(A)f Fo(\000)-7 b(!)12 b Fn(A)7 b Fo(\002)g Fn(B)17 b Fe(b)n(e)18 2370 y(the)h(morphism)g(with)g Fn(p)431 2377 y Fk(A)467 2357 y Fm(e)458 2370 y Fn(f)i Fu(=)15 b(id)j Fe(and)g Fn(p)718 2377 y Fk(B)756 2357 y Fm(e)746 2370 y Fn(f)i Fu(=)15 b Fn(f)5 b Fe(.)25 b(Then)16 b Fu(\()p Fn(A;)1105 2357 y Fm(e)1095 2370 y Fn(f)5 b Fu(\))18 b Fe(is)f(a)g(sub)n(obje)n(ct)g (of)h Fn(A)11 b Fo(\002)g Fn(B)r Fe(,)18 2424 y(namely)16 b(the)g(gr)n(aph)h(of)f Fn(f)5 b Fe(.)611 2738 y Ff(krs.tex)23 b(-)g(Date:)47 b(June)24 b(23,)f(1994)g(Time:)47 b(10:45)p eop %%Page: 6 6 6 5 bop 18 54 a Fu(6)533 b Fh(Bo)q(do)11 b(P)o(areigis)f(and)g(Helm)o (ut)h(R\177)-18 b(ohrl)18 145 y Fv(Prop)q(osition)18 b(2.6)g Fe(L)n(et)h Fn(\023)483 152 y Fk(X)533 145 y Fu(:)f Fn(X)j Fo(\000)-7 b(!)18 b Fn(A)12 b Fo(\002)h Fn(B)21 b Fe(b)n(e)e(a)g(c)n(omplement)g(of)h Fn(B)r Fe(.)29 b(Then)19 b(we)g(have)18 199 y Fn(X)13 b Fo(\\)e Fn(B)j Fu(=)f(0)j Fe(and)g Fn(X)e Fo([)c Fn(B)k Fu(=)f Fn(A)d Fo(\002)h Fn(B)r Fe(.)76 282 y Fd(Pro)q(of:)65 b Fu(Since)23 b(a)e(complemen)o(t)h(is)h(a)e(w)o(eak)g(complemen)o(t)h (w)o(e)f(get)h Fn(X)17 b Fo(\\)e Fn(B)25 b Fu(=)f(0)d(b)o(y)18 336 y(Prop)q(osition)16 b(2.3.)i(In)e(the)g(comm)o(utativ)o(e)e (diagram)572 408 y Fn(A)147 b(A)795 376 y Fm(`)842 408 y Fn(B)p 619 398 121 2 v 698 397 a Fl(-)1024 408 y Fn(B)p 892 398 120 2 v 892 397 a Fl(\033)568 636 y Fn(X)d(X)798 604 y Fm(`)845 636 y Fn(B)p 623 625 114 2 v 695 624 a Fl(-)659 610 y Fk(j)673 614 y Fj(X)1024 636 y Fn(B)p 895 625 116 2 v 895 624 a Fl(\033)572 865 y Fn(A)j(A)10 b Fo(\002)g Fn(B)p 619 852 120 2 v 619 851 a Fl(\033)659 837 y Fk(p)676 841 y Fj(A)1024 865 y Fn(B)p 892 852 119 2 v 969 851 a Fl(-)931 837 y Fk(p)948 841 y Fj(B)p 588 592 2 164 v 589 592 a Fl(?)607 512 y Fg(\030)607 521 y Fi(=)p 588 820 V 589 820 a Fl(?)607 739 y Fg(\030)607 748 y Fi(=)p 815 592 V 816 592 a Fl(?)834 512 y Fg(\030)834 521 y Fi(=)p 815 820 V 816 820 a Fl(?)p 1042 592 V 1043 592 a(?)1061 516 y Fi(=)p 1042 820 V 1043 820 a Fl(?)1061 743 y Fi(=)730 742 y Fk(\023)742 746 y Fj(X)620 698 y Fl(@)662 739 y(@)703 781 y(@)742 820 y(@)-42 b(R)970 698 y(\000)928 739 y(\000)887 781 y(\000)848 820 y(\000)g(\011)18 933 y Fu(the)19 b(morphism)g Fn(A)358 901 y Fm(`)404 933 y Fn(B)h Fo(\000)-7 b(!)18 b Fn(A)13 b Fo(\002)g Fn(B)20 b Fu(is)f(a)g(di\013erence)h(cok)o(ernel)f(b)o(y)g(axiom)f (\(I\))h(and)g(so)f(is)18 987 y Fn(X)67 955 y Fm(`)113 987 y Fn(B)d Fo(\000)-7 b(!)12 b Fn(A)e Fo(\002)g Fn(B)r Fu(.)20 b(Hence)c Fn(X)d Fo([)e Fn(B)j Fu(=)f Fn(A)d Fo(\002)h Fn(B)r Fu(.)p 1613 960 24 2 v 1613 985 2 25 v 1635 985 V 1613 987 24 2 v 18 1141 a Fv(Theorem)17 b(2.7)53 b Fe(1.)19 b Fn(A)7 b Fo(\002)g Fn(B)17 b Fe(is)e(an)g (internal)f(dir)n(e)n(ct)g(pr)n(o)n(duct)i(of)f(the)g(sub)n(obje)n(cts) f Fn(A)h Fe(and)g Fn(B)r Fe(.)53 1226 y(2.)k(If)c Fn(A)h Fe(and)f Fn(B)j Fe(ar)n(e)d(sub)n(obje)n(cts)g(of)h Fn(U)k Fe(such)c(that)g Fn(U)21 b Fe(is)15 b(an)h(internal)e(dir)n(e)n(ct)i (pr)n(o)n(duct)g(of)g Fn(A)109 1280 y Fe(and)g Fn(B)r Fe(,)g(then)g(ther)n(e)g(is)g(an)g(isomorphism)h Fn(U)909 1267 y Fo(\030)909 1282 y Fu(=)956 1280 y Fn(A)11 b Fo(\002)f Fn(B)18 b Fe(such)f(that)854 1352 y Fn(U)628 1465 y(A)709 1418 y Fl(\010)750 1397 y(\010)767 1389 y(\010)-42 b(*)1081 1465 y Fn(B)994 1418 y Fl(H)953 1397 y(H)936 1389 y(H)g(Y)p 871 1531 2 164 v 872 1531 a(?)890 1451 y Fg(\030)890 1460 y Fi(=)809 1577 y Fn(A)10 b Fo(\002)h Fn(B)709 1502 y Fl(H)750 1523 y(H)767 1531 y(H)-42 b(j)994 1502 y(\010)953 1523 y(\010)936 1531 y(\010)g(\031)109 1645 y Fe(c)n(ommutes.)76 1728 y Fd(Pro)q(of:)57 b Fu(\(1\))18 b(Since)j Fn(p)479 1735 y Fk(A)505 1726 y Fm(e)506 1728 y Fn(\023)522 1735 y Fk(A)569 1728 y Fu(=)f(id)g(is)g(an)g(isomorphism,)h Fn(A)e Fu(is)h(a)f(complemen)o(t)i(of)e Fn(B)i Fu(in)18 1782 y Fn(A)10 b Fo(\002)g Fn(B)r Fu(.)19 b(Th)o(us)c(b)o(y)g(Lemma)g (2.2)f(and)h(Prop)q(osition)h(2.6)e(w)o(e)g(get)h(that)f Fn(A)c Fo(\002)g Fn(B)16 b Fu(is)g(an)f(in)o(ternal)18 1836 y(direct)h(pro)q(duct)f(of)g Fn(A)g Fu(and)h Fn(B)r Fu(.)76 1890 y(\(2\))k(Giv)o(en)i(an)f(in)o(ternal)h(direct)g(pro)q (duct)g Fn(U)k Fu(of)21 b Fn(A)h Fu(and)f Fn(B)r Fu(.)38 b(Let)21 b Fn(U)28 b Fo(\000)-7 b(!)22 b Fn(X)j Fu(b)q(e)d(the)18 1943 y(cok)o(ernel)14 b(of)e Fn(A)h Fo(\000)-7 b(!)12 b Fn(U)5 b Fu(.)19 b(Then)14 b Fn(A)e Fo(\000)-7 b(!)12 b Fn(U)18 b Fu(is)c(the)f(k)o(ernel)h(of)f Fn(U)k Fo(\000)-7 b(!)12 b Fn(X)17 b Fu(since)d(it)f(w)o(as)g(a)f(k)o(ernel.)18 1997 y(In)k(the)f(comm)o(utativ)o(e)f(diagram)744 2258 y(0)152 b Fn(B)p 781 2243 125 2 v 864 2242 a Fl(-)738 2440 y Fn(A)147 b(U)p 786 2425 120 2 v 864 2424 a Fl(-)1098 2440 y Fn(X)p 969 2425 117 2 v 1044 2424 a Fl(-)515 2072 y Fu(Ke\()p Fn(f)5 b Fu(\))637 2113 y Fl(H)679 2134 y(H)720 2155 y(H)762 2176 y(H)803 2196 y(H)832 2211 y(H)-42 b(j)606 2134 y(@)647 2176 y(@)682 2211 y(@)g(R)590 2134 y(A)610 2176 y(A)631 2217 y(A)652 2259 y(A)673 2300 y(A)693 2342 y(A)714 2383 y(A)719 2392 y(A)-21 b(U)p 755 2392 2 119 v 16 w(?)p 936 2392 V 139 w(?)1056 2341 y Fk(f)969 2316 y Fl(@)1011 2357 y(@)1046 2392 y(@)-42 b(R)611 2738 y Ff(krs.tex)23 b(-)g(Date:)47 b(June)24 b(23,)f(1994)g(Time:)47 b(10:45)p eop %%Page: 7 7 7 6 bop 644 54 a Fh(Krull-Sc)o(hmid)o(t)9 b(Theorem)601 b Fu(7)18 145 y(the)16 b(morphism)h(Ke\()p Fn(f)5 b Fu(\))14 b Fo(\000)-7 b(!)14 b Fn(A)i Fu(and)h(Ke\()p Fn(f)5 b Fu(\))14 b Fo(\000)-7 b(!)14 b Fu(0)i(exist)g(since)i Fn(A)c Fo(\000)-7 b(!)14 b Fn(U)21 b Fu(is)c(the)f(k)o(ernel)h(of)18 199 y Fn(U)g Fo(\000)-7 b(!)12 b Fn(X)19 b Fu(and)c(0)d(=)h Fn(A)e Fo(\\)f Fn(B)17 b Fu(is)f(a)e(pull-bac)o(k.)22 b(Hence)16 b(Ke)q(\()p Fn(f)5 b Fu(\))12 b(=)h(0.)19 b(No)o(w)c(let)718 315 y Fn(P)152 b(U)p 767 301 120 2 v 845 300 a Fl(-)818 286 y Fk(g)717 497 y Fn(B)144 b(X)p 767 483 116 2 v 841 482 a Fl(-)815 467 y Fk(f)p 734 450 2 119 v 735 450 a Fl(?)p 916 450 V 140 w(?)18 594 y Fu(b)q(e)16 b(a)f(pull-bac)o(k)i(and)e(consider)h(the)f(comm)o(utativ)o(e)g (diagrams)73 875 y(\(*\))557 879 y Fn(P)152 b(U)p 606 864 120 2 v 684 863 a Fl(-)657 849 y Fk(g)556 1061 y Fn(B)145 b(X)p 606 1046 116 2 v 680 1045 a Fl(-)654 1031 y Fk(f)375 697 y Fn(B)604 777 y Fk(\023)616 781 y Fj(B)457 734 y Fl(H)498 755 y(H)540 776 y(H)581 797 y(H)623 817 y(H)651 832 y(H)-42 b(j)425 755 y(@)466 797 y(@)501 832 y(@)g(R)428 874 y Fi(id)409 755 y Fl(A)430 797 y(A)450 838 y(A)471 880 y(A)492 921 y(A)513 963 y(A)533 1004 y(A)538 1013 y(A)-21 b(U)p 574 1013 2 119 v 16 w(?)p 756 1013 V 139 w(?)1049 879 y Fn(P)152 b(U)p 1098 864 120 2 v 1176 863 a Fl(-)1149 849 y Fk(g)1048 1061 y Fn(B)146 b(X)q(:)p 1098 1046 105 2 v 1161 1045 a Fl(-)1141 1031 y Fk(f)868 697 y Fn(A)1095 777 y Fk(\023)1107 781 y Fj(A)948 734 y Fl(H)990 755 y(H)1031 776 y(H)1073 797 y(H)1114 817 y(H)1143 832 y(H)-42 b(j)916 755 y(@)958 797 y(@)993 832 y(@)g(R)931 873 y Fi(0)900 755 y Fl(A)921 797 y(A)942 838 y(A)963 880 y(A)983 921 y(A)1004 963 y(A)1025 1004 y(A)1030 1013 y(A)-21 b(U)p 1065 1013 2 119 v 15 w(?)p 1247 1013 V 140 w(?)76 1158 y Fu(They)14 b(induce)h(the)f(canonical)h (morphism)f Fn(A)862 1126 y Fm(`)908 1158 y Fn(B)g Fo(\000)-7 b(!)12 b Fn(P)19 b Fo(\000)-7 b(!)12 b Fn(U)5 b Fu(,)14 b(whic)o(h)g(is)g(a)g(di\013erence)18 1212 y(cok)o(ernel.)20 b(Since)d(Ke)q(\()p Fn(f)5 b Fu(\))12 b(=)h(0)i(the)g(diagram)867 1506 y Fn(P)153 b(U)p 916 1492 120 2 v 994 1491 a Fl(-)967 1476 y Fk(g)867 1688 y Fn(B)144 b(X)p 917 1673 116 2 v 991 1672 a Fl(-)965 1658 y Fk(f)646 1320 y Fu(Ke)q(\()p Fn(g)r Fu(\))767 1362 y Fl(H)808 1382 y(H)850 1403 y(H)891 1424 y(H)933 1445 y(H)962 1459 y(H)-42 b(j)735 1382 y(@)777 1424 y(@)812 1459 y(@)g(R)p 702 1641 2 300 v 703 1641 a(?)p 884 1641 2 119 v 140 w(?)p 1066 1641 V 140 w(?)567 1684 y Fu(0)13 b(=)g(Ke\()p Fn(f)5 b Fu(\))p 776 1673 78 2 v 812 1672 a Fl(-)18 1792 y Fu(comm)o(utes)18 b(and)h(the)g (canonical)h(morphism)g(Ke\()p Fn(g)r Fu(\))e Fo(\000)-8 b(!)19 b Fn(P)25 b Fu(is)19 b(the)g(zero)g(morphism.)32 b(By)18 1846 y(axiom)15 b(\(I)q(I\))h Fn(g)g Fu(is)g(an)f(isomorphism.) 21 b(Th)o(us)15 b(w)o(e)g(can)h(replace)g Fn(P)22 b Fu(b)o(y)15 b Fn(U)20 b Fu(and)c Fn(g)g Fu(b)o(y)f(id)i(so)d(that)18 1913 y(there)20 b(is)h(a)f(factorization)g Fn(U)26 b Fo(\000)-8 b(!)21 b Fn(B)737 1886 y Fk(f)711 1913 y Fo(\000)-8 b(!)21 b Fn(X)i Fu(of)d Fn(U)26 b Fo(\000)-7 b(!)20 b Fn(X)t Fu(,)g(whic)o(h)h(is)g(the)f(cok)o(ernel)h(of)18 1967 y Fn(A)13 b Fo(\000)-8 b(!)12 b Fn(U)5 b Fu(.)76 2021 y(Since)21 b(\()p Fn(A)f Fo(\000)-7 b(!)19 b Fn(U)25 b Fo(\000)-7 b(!)19 b Fn(X)t Fu(\))g(=)i(0)e(and)h(Ke\()p Fn(f)5 b Fu(\))20 b(=)g(0)f(w)o(e)h(get)f(\()p Fn(A)h Fo(\000)-7 b(!)19 b Fn(U)25 b Fo(\000)-7 b(!)19 b Fn(B)r Fu(\))h(=)g(0)18 2075 y(and)g(hence)h(a)e(factorization)g(\()p Fn(U)25 b Fo(\000)-7 b(!)20 b Fn(X)j Fo(\000)-7 b(!)19 b Fn(B)r Fu(\))h(=)h(\()p Fn(U)j Fo(\000)-7 b(!)20 b Fn(B)r Fu(\).)33 b(Since)21 b Fn(U)k Fo(\000)-7 b(!)19 b Fn(X)k Fu(is)18 2142 y(the)17 b(cok)o(ernel)i(of)e Fn(A)f Fo(\000)-7 b(!)16 b Fn(U)22 b Fu(w)o(e)c(get)f(\()p Fn(X)i Fo(\000)-7 b(!)16 b Fn(B)918 2115 y Fk(f)891 2142 y Fo(\000)-7 b(!)16 b Fn(X)t Fu(\))f(=)i(id)q(.)27 b(No)o(w)17 b Fn(f)22 b Fu(has)c(k)o(ernel)g(zero)18 2208 y(and)j(is)h(a)f (di\013erence)h(cok)o(ernel)g(of)e(\()p Fn(B)761 2181 y Fk(f)734 2208 y Fo(\000)-7 b(!)22 b Fn(X)k Fo(\000)-7 b(!)22 b Fn(B)r(;)8 b(B)1129 2183 y Fi(id)1106 2208 y Fo(\000)-7 b(!)22 b Fn(B)r Fu(\).)37 b(Hence)22 b(b)o(y)f(axiom)18 2262 y(\(I)q(I\))e(w)o(e)f(obtain)h(that)f Fn(f)24 b Fu(:)18 b Fn(B)i Fo(\000)-7 b(!)17 b Fn(X)22 b Fu(is)d(an)g (isomorphism.)31 b(Therefore)18 b(w)o(e)h(ma)o(y)f(replace)18 2316 y Fn(X)23 b Fu(b)o(y)c Fn(B)i Fu(and)f(consider)g Fn(U)25 b Fo(\000)-7 b(!)19 b Fn(B)j Fu(as)d(cok)o(ernel)h(of)f Fn(A)h Fo(\000)-7 b(!)19 b Fn(U)5 b Fu(.)33 b(F)l(urthermore)19 b(w)o(e)g(ha)o(v)o(e)18 2370 y(\()p Fn(B)14 b Fo(\000)-7 b(!)12 b Fn(U)17 b Fo(\000)-7 b(!)12 b Fn(B)r Fu(\))g(=)h(id)k(b)o(y)e (diagram)g(\()p Fo(\003)p Fu(\).)k(Analogously)c(w)o(e)g(get)g(a)g (morphism)h Fn(U)h Fo(\000)-7 b(!)12 b Fn(A)p Fu(,)18 2424 y(whic)o(h)k(is)g(a)e(cok)o(ernel)i(of)f Fn(B)g Fo(\000)-8 b(!)12 b Fn(U)5 b Fu(,)15 b(suc)o(h)h(that)e(\()p Fn(A)f Fo(\000)-8 b(!)12 b Fn(U)18 b Fo(\000)-7 b(!)12 b Fn(A)p Fu(\))g(=)h(id)q(.)611 2738 y Ff(krs.tex)23 b(-)g(Date:)47 b(June)24 b(23,)f(1994)g(Time:)47 b(10:45)p eop %%Page: 8 8 8 7 bop 18 54 a Fu(8)533 b Fh(Bo)q(do)11 b(P)o(areigis)f(and)g(Helm)o (ut)h(R\177)-18 b(ohrl)76 145 y Fu(No)o(w)11 b(w)o(e)g(pro)o(v)o(e)f Fn(U)406 133 y Fo(\030)406 147 y Fu(=)454 145 y Fn(A)r Fo(\002)r Fn(B)r Fu(.)19 b(Let)12 b Fn(Y)21 b Fu(b)q(e)12 b(the)g(k)o(ernel)g(of)f Fn(h)g Fu(in)h(the)f(follo)o(wing)h(comm)o (utativ)o(e)18 199 y(diagram)570 243 y Fn(A)192 b(Y)p 617 229 165 2 v 617 228 a Fl(\033)1023 243 y Fn(B)p 846 229 164 2 v 968 228 a Fl(-)570 470 y Fn(A)g(U)p 617 456 166 2 v 617 455 a Fl(\033)1023 470 y Fn(B)p 845 456 164 2 v 967 455 a Fl(-)570 696 y Fn(A)147 b(A)10 b Fo(\002)g Fn(B)p 617 683 120 2 v 617 682 a Fl(\033)1023 696 y Fn(B)p 891 683 119 2 v 968 682 a Fl(-)p 586 423 2 164 v 587 423 a(?)605 347 y Fi(=)p 813 423 V 814 423 a Fl(?)p 1040 423 V 185 w(?)1059 347 y Fi(=)p 586 650 V 587 650 a Fl(?)605 574 y Fi(=)p 813 650 V 814 650 a Fl(?)832 578 y Fk(h)p 1040 650 V 1041 650 a Fl(?)1059 574 y Fi(=)619 301 y Fl(@)660 343 y(@)702 384 y(@)741 423 y(@)-42 b(R)968 301 y(\000)926 343 y(\000)885 384 y(\000)846 423 y(\000)g(\011)18 753 y Fu(T)l(o)14 b(pro)o(v)o(e)h(the)g(existence)h(of)f Fn(Y)23 b Fo(\000)-8 b(!)12 b Fn(A)j Fu(observ)o(e)g(that)f Fn(A)f Fo(\000)-7 b(!)12 b Fn(U)20 b Fu(is)15 b(the)g(k)o(ernel)h(of)f Fn(U)i Fo(\000)-7 b(!)12 b Fn(B)18 807 y Fu(and)17 b(that)g(\()p Fn(Y)26 b Fo(\000)-7 b(!)15 b Fn(U)21 b Fo(\000)-7 b(!)16 b Fn(B)r Fu(\))f(=)h(\()p Fn(Y)27 b Fo(\000)-8 b(!)16 b Fn(U)21 b Fo(\000)-7 b(!)15 b Fn(A)d Fo(\002)f Fn(B)18 b Fo(\000)-7 b(!)16 b Fn(B)r Fu(\))f(=)h(0.)26 b(Th)o(us)17 b(w)o(e)g(get)g(a)18 861 y(comm)o(utativ)o(e)d(diagram)816 1086 y(0)153 b Fn(A)p 853 1072 127 2 v 938 1071 a Fl(-)809 1268 y Fn(B)147 b(U)p 859 1253 119 2 v 936 1252 a Fl(-)628 905 y Fn(Y)709 942 y Fl(H)751 962 y(H)792 983 y(H)834 1004 y(H)875 1025 y(H)904 1039 y(H)-42 b(j)678 962 y(@)719 1004 y(@)754 1039 y(@)g(R)662 962 y(A)682 1004 y(A)703 1045 y(A)724 1087 y(A)745 1128 y(A)765 1170 y(A)786 1211 y(A)791 1221 y(A)-21 b(U)p 827 1221 2 119 v 16 w(?)p 1008 1221 V 139 w(?)18 1323 y Fu(as)15 b Fn(A)c Fo(\\)f Fn(B)15 b Fu(=)f(0.)21 b(This)16 b(sho)o(ws)g(that)e(\()p Fn(Y)24 b Fo(\000)-7 b(!)13 b Fn(U)5 b Fu(\))13 b(=)h(0.)20 b(Th)o(us)c Fn(h)e Fu(:)f Fn(U)18 b Fo(\000)-7 b(!)13 b Fn(A)d Fo(\002)h Fn(B)17 b Fu(has)f(k)o(ernel)18 1377 y(0.)j(On)d(the)f(other)g(hand)h(w)o(e)f(ha)o(v)o(e)g(a)f(comm)o (utativ)o(e)h(diagram)570 1678 y Fn(A)192 b(U)p 617 1663 166 2 v 741 1662 a Fl(-)1023 1678 y Fn(B)p 845 1663 164 2 v 845 1662 a Fl(\033)570 1903 y Fn(A)147 b(A)10 b Fo(\002)g Fn(B)p 617 1890 120 2 v 617 1889 a Fl(\033)1023 1903 y Fn(B)p 891 1890 119 2 v 968 1889 a Fl(-)751 1446 y Fn(A)793 1414 y Fm(`)840 1446 y Fn(B)619 1630 y Fl(\000)660 1589 y(\000)702 1547 y(\000)741 1508 y(\000)-42 b(\022)968 1630 y(@)926 1589 y(@)885 1547 y(@)846 1508 y(@)g(I)619 1735 y(@)660 1777 y(@)702 1818 y(@)741 1857 y(@)g(R)968 1735 y(\000)926 1777 y(\000)885 1818 y(\000)846 1857 y(\000)g(\011)p 586 1857 2 164 v -301 w(?)605 1781 y Fi(=)p 813 1857 V 814 1857 a Fl(?)832 1786 y Fk(h)p 1040 1857 V 1041 1857 a Fl(?)1059 1781 y Fi(=)p 813 1630 V 814 1630 a Fl(?)18 1968 y Fu(whic)o(h)16 b(implies)i(b)o(y)e(axiom)g (\(I\))f(that)g Fn(A)711 1936 y Fm(`)758 1968 y Fn(B)g Fo(\000)-7 b(!)13 b Fn(U)18 b Fo(\000)-7 b(!)13 b Fn(A)d Fo(\002)h Fn(B)18 b Fu(is)e(a)f(di\013erence)i(cok)o(ernel.)18 2022 y(By)12 b(axiom)h(\(I)q(I\))f Fn(h)h Fu(is)g(an)f(isomorphism)h (and)g(the)f(ab)q(o)o(v)o(e)g(diagram)g(pro)o(v)o(es)g(that)g(the)g (diagram)18 2076 y(in)k(the)f(theorem)g(comm)o(utes.)p 1613 2049 24 2 v 1613 2074 2 25 v 1635 2074 V 1613 2076 24 2 v 536 2229 a Fv(3)52 b(Summable)18 b(morphisms)18 2316 y Fu(In)c(this)f(paragraph)g(w)o(e)g(shall)h(in)o(tro)q(duce)g(an) g(addition)g(of)f(certain)g(morphisms.)20 b(One)14 b(of)f(the)18 2370 y(aims)h(is)h(the)f(pro)q(of)g(of)g(a)g(form)o(ula)f(id)h(=)714 2368 y Fm(e)715 2370 y Fn(\023)731 2377 y Fk(A)758 2370 y Fn(p)781 2377 y Fk(A)816 2370 y Fu(+)858 2368 y Fm(e)860 2370 y Fn(\023)876 2377 y Fk(B)904 2370 y Fn(p)927 2377 y Fk(B)970 2370 y Fu(for)f Fn(A)c Fo(\002)f Fn(B)r Fu(.)19 b(Let)c Fo(C)h Fu(b)q(e)f(as)f(in)h(sections)18 2424 y(1)g(and)g(2.)611 2738 y Ff(krs.tex)23 b(-)g(Date:)47 b(June)24 b(23,)f(1994)g(Time:)47 b(10:45)p eop %%Page: 9 9 9 8 bop 644 54 a Fh(Krull-Sc)o(hmid)o(t)9 b(Theorem)601 b Fu(9)18 145 y Fv(De\014nition)19 b(3.1)f Fu(Let)f Fn(\033)g Fu(:)e Fn(A)543 113 y Fm(`)590 145 y Fn(A)h Fo(\000)-8 b(!)15 b Fn(A)d Fo(\002)f Fn(A)17 b Fu(b)q(e)h(the)f(canonical)h (morphism)f(de\014ned)i(b)o(y)18 199 y(the)c(comm)o(utativ)o(e)g (diagram)583 291 y Fn(A)149 b(A)808 259 y Fm(`)854 291 y Fn(A)p 631 281 123 2 v 712 280 a Fl(-)676 265 y Fk(j)690 269 y Fb(1)1037 291 y Fn(A)p 902 281 V 902 280 a Fl(\033)948 265 y Fk(j)962 269 y Fb(2)583 520 y Fn(A)f(A)11 b Fo(\002)f Fn(A)p 631 508 122 2 v 631 507 a Fl(\033)674 528 y Fk(p)691 532 y Fb(1)1037 520 y Fn(A)p 903 508 V 983 507 a Fl(-)946 528 y Fk(p)963 532 y Fb(2)p 599 475 2 164 v 600 475 a Fl(?)554 403 y Fi(id)p 1054 475 V 1054 475 a Fl(?)1073 403 y Fi(id)p 826 475 V 827 475 a Fl(?)664 332 y(H)705 353 y(H)747 374 y(H)788 394 y(H)830 415 y(H)871 436 y(H)913 457 y(H)949 475 y(H)-42 b(j)949 332 y(\010)908 353 y(\010)866 374 y(\010)825 394 y(\010)783 415 y(\010)742 436 y(\010)700 457 y(\010)664 475 y(\010)g(\031)845 456 y Fk(\033)725 431 y Fi(0)171 b(0)18 605 y Fu(By)19 b(axiom)f(\(I\))g Fn(\033)i Fu(is)f(a)g(di\013erence)g(cok)o(ernel.)31 b(Let)19 b Fn(f)r(;)8 b(g)19 b Fo(2)g(C)s Fu(\()p Fn(A;)8 b(B)r Fu(\).)28 b Fn(f)23 b Fu(and)c Fn(g)h Fu(are)e(called)18 659 y Fe(summable)d Fu(if)h(the)f(canonical)h(morphism)g Fn(h)f Fu(of)583 738 y Fn(A)149 b(A)808 706 y Fm(`)854 738 y Fn(A)p 631 727 123 2 v 712 726 a Fl(-)1037 738 y Fn(A)p 902 727 V 902 726 a Fl(\033)809 856 y Fn(B)665 790 y Fk(f)664 779 y Fl(H)705 800 y(H)722 808 y(H)-42 b(j)p 826 808 2 50 v 63 w(?)845 793 y Fk(h)970 787 y(g)949 779 y Fl(\010)908 800 y(\010)891 808 y(\010)g(\031)18 928 y Fu(factors)11 b(\(necessarily)i(uniquely\))h(through)d Fn(\033)r Fu(.)19 b(The)12 b(factorization)f(morphism)i Fn(A)t Fo(\002)t Fn(A)f Fo(\000)-7 b(!)12 b Fn(B)18 982 y Fu(will)17 b(b)q(e)f(written)f(as)g Fo(h)p Fn(f)r(;)8 b(g)r Fo(i)p Fu(.)76 1036 y(A)14 b(similar)i(de\014nition)g(can)f(b)q (e)g(giv)o(en)g(for)f Fn(n)h Fu(morphisms.)20 b(The)14 b(family)h(\()p Fn(f)1365 1043 y Fk(i)1379 1036 y Fo(j)p Fn(i)d Fu(=)h(1)p Fn(;)8 b(:)g(:)g(:)d(;)j(n)p Fu(\))18 1090 y(is)22 b Fe(summable)g Fu(if)g(for)f(all)i Fn(i;)8 b(j)23 b Fu(with)f(1)h Fo(\024)h Fn(i)f(<)h(j)i Fo(\024)d Fn(n)f Fu(the)g(morphism)g(induced)i(b)o(y)e(the)18 1144 y Fn(f)40 1151 y Fk(i)54 1144 y Fn(;)8 b(f)97 1151 y Fk(i)p Fi(+1)152 1144 y Fn(;)g(:)g(:)g(:)d(;)j(f)276 1151 y Fk(j)311 1144 y Fu(factors)18 b(through)638 1112 y Fm(`)677 1122 y Fk(j)677 1155 y(k)q Fi(=)p Fk(i)742 1144 y Fn(A)h Fo(\000)-8 b(!)886 1112 y Fm(Q)925 1122 y Fk(j)925 1155 y(k)q Fi(=)p Fk(i)990 1144 y Fn(A)p Fu(.)30 b(The)19 b(factorization)f(is)h(denoted)18 1198 y(b)o(y)c Fo(h)p Fn(f)121 1205 y Fk(i)135 1198 y Fn(;)8 b(f)178 1205 y Fk(i)p Fi(+1)233 1198 y Fn(;)g(:)g(:)g(:)d(;)j(f)357 1205 y Fk(j)374 1198 y Fo(i)k Fu(:)g Fn(A)f Fo(\002)f Fn(:)e(:)g(:)g Fo(\002)j Fn(A)h Fo(\000)-7 b(!)12 b Fn(B)r Fu(.)76 1252 y(If)19 b(\()p Fn(f)165 1259 y Fk(i)179 1252 y Fo(j)p Fn(i)f Fu(=)i(1)p Fn(;)8 b(:)g(:)g(:)t(;)g(n)p Fu(\))18 b(is)i(summable)g(then)f(the)g Fe(sum)g Fn(f)1047 1259 y Fi(1)1079 1252 y Fu(+)13 b Fn(:)8 b(:)g(:)i Fu(+)j Fn(f)1262 1259 y Fk(n)1304 1252 y Fu(is)20 b(de\014ned)g(as)f(the)18 1322 y(morphism)c Fn(A)290 1297 y Fi(\001)317 1284 y Fj(n)277 1322 y Fo(\000)-7 b(!)12 b Fn(A)e Fo(\002)h Fn(:)d(:)g(:)g Fo(\002)i Fn(A)608 1293 y Fg(h)p Fk(f)637 1297 y Fb(1)654 1293 y Fk(;:::)o(;f)719 1297 y Fj(n)740 1293 y Fg(i)644 1322 y Fo(\000)-7 b(!)48 b Fn(B)r(:)18 1412 y Fv(Lemma)17 b(3.2)h Fe(L)n(et)g Fn(f)391 1419 y Fk(i)423 1412 y Fo(2)g(C)s Fu(\()p Fn(A;)8 b(B)r Fu(\))p Fe(,)18 b Fn(i)f Fu(=)h(1)p Fn(;)8 b(:)g(:)g(:)d(;)j(n)p Fe(,)19 b Fn(h)f Fo(2)g(C)s Fu(\()p Fn(B)r(;)8 b(C)s Fu(\))16 b Fe(and)j Fn(k)g Fo(2)f(C)s Fu(\()p Fn(D)q(;)8 b(A)p Fu(\))p Fe(.)28 b(L)n(et)18 1465 y Fu(\()p Fn(f)58 1472 y Fk(i)72 1465 y Fo(j)p Fn(i)14 b Fu(=)i(1)p Fn(;)8 b(:)g(:)g(:)d(;)j(n)p Fu(\))17 b Fe(b)n(e)g(summable.)26 b(Then)17 b Fu(\()p Fn(f)804 1472 y Fk(i)818 1465 y Fn(k)q Fo(j)p Fn(i)e Fu(=)h(1)p Fn(;)8 b(:)g(:)g(:)t(;)g(n)p Fu(\))18 b Fe(and)f Fu(\()p Fn(hf)1280 1472 y Fk(i)1294 1465 y Fo(j)p Fn(i)e Fu(=)h(1)p Fn(;)8 b(:)g(:)g(:)d(;)j(n)p Fu(\))17 b Fe(ar)n(e)18 1519 y(summable)g(and)f(we)h(have)h Fn(h)p Fu(\()p Fn(f)557 1526 y Fi(1)586 1519 y Fu(+)11 b Fn(:)d(:)g(:)g Fu(+)j Fn(f)763 1526 y Fk(n)786 1519 y Fu(\))i(=)h Fn(hf)914 1526 y Fi(1)944 1519 y Fu(+)d Fn(:)d(:)g(:)g Fu(+)j Fn(hf)1147 1526 y Fk(n)1187 1519 y Fe(and)17 b Fu(\()p Fn(f)1316 1526 y Fi(1)1345 1519 y Fu(+)11 b Fn(:)d(:)g(:)g Fu(+)j Fn(f)1522 1526 y Fk(n)1545 1519 y Fu(\))p Fn(k)j Fu(=)18 1573 y Fn(f)40 1580 y Fi(1)59 1573 y Fn(k)d Fu(+)f Fn(:)e(:)g(:)g Fu(+)j Fn(f)270 1580 y Fk(n)293 1573 y Fn(k)q(:)76 1663 y Fd(Pro)q(of:)96 b Fu(It)30 b(is)g(su\016cien)o(t)g(to)e(pro)o(v)o(e)h(that)g (factorizations)g Fo(h)p Fn(hf)1305 1670 y Fi(1)1324 1663 y Fn(;)8 b(:)g(:)g(:)d(;)j(hf)1474 1670 y Fk(n)1496 1663 y Fo(i)29 b Fu(resp.)18 1717 y Fo(h)p Fn(f)58 1724 y Fi(1)76 1717 y Fn(k)q(;)8 b(:)g(:)g(:)d(;)j(f)225 1724 y Fk(n)247 1717 y Fn(k)q Fo(i)18 b Fu(exist.)30 b(Observ)o(e)18 b(that)g(the)g(factorization)g Fo(h)p Fn(f)1110 1724 y Fi(1)1129 1717 y Fn(;)8 b(:)g(:)g(:)t(;)g(f)1252 1724 y Fk(n)1275 1717 y Fo(i)17 b Fu(is)i(the)f(only)h(mor-)18 1771 y(phism)d(whic)o(h)g(mak)o(es)f(all)h(the)f(diagrams)624 1876 y Fn(A)181 b(A)873 1860 y Fk(n)p 671 1862 155 2 v 784 1861 a Fl(-)732 1848 y Fm(e)-19 b Fk(\023)748 1852 y Fj(i)696 1925 y Fk(f)712 1929 y Fj(i)704 1913 y Fl(H)746 1934 y(H)763 1943 y(H)-42 b(j)849 1990 y Fn(B)p 867 1943 2 50 v 868 1943 a Fl(?)886 1925 y Fg(h)p Fk(f)915 1929 y Fb(1)932 1925 y Fk(;:::)o(;f)997 1929 y Fj(n)1018 1925 y Fg(i)18 2062 y Fu(comm)o(ute.)19 b(Th)o(us)c(the)h(diagrams)536 2167 y Fn(A)228 b(A)832 2151 y Fk(n)p 584 2153 201 2 v 743 2152 a Fl(-)667 2139 y Fm(e)-19 b Fk(\023)683 2143 y Fj(i)1081 2167 y Fn(A)p 868 2153 V 868 2152 a Fl(\033)944 2143 y Fi(\001)971 2130 y Fj(n)808 2303 y Fn(B)719 2227 y Fk(f)735 2231 y Fj(i)617 2204 y Fl(H)658 2225 y(H)700 2246 y(H)721 2256 y(H)-42 b(j)993 2204 y(\010)952 2225 y(\010)910 2246 y(\010)890 2256 y(\010)g(\031)969 2238 y Fk(f)985 2242 y Fb(1)1001 2238 y Fi(+)p Fk(:::)p Fi(+)p Fk(f)1097 2242 y Fj(n)808 2440 y Fn(C)613 2295 y Fk(hf)648 2299 y Fj(i)585 2225 y Fl(@)627 2266 y(@)668 2308 y(@)710 2350 y(@)751 2391 y(@)753 2392 y(@)g(R)990 2291 y Fk(g)1025 2225 y Fl(\000)984 2266 y(\000)942 2308 y(\000)901 2350 y(\000)859 2391 y(\000)858 2392 y(\000)g(\011)p 825 2256 2 73 v 826 2256 a(?)844 2204 y Fg(h)p Fk(f)873 2208 y Fb(1)890 2204 y Fk(;:::)o(;f)955 2208 y Fj(n)976 2204 y Fg(i)p 825 2392 V 826 2392 a Fl(?)844 2366 y Fk(h)611 2738 y Ff(krs.tex)23 b(-)g(Date:)47 b(June)24 b(23,)f(1994)g(Time:)47 b(10:45)p eop %%Page: 10 10 10 9 bop 18 54 a Fu(10)510 b Fh(Bo)q(do)11 b(P)o(areigis)f(and)g(Helm)o (ut)h(R\177)-18 b(ohrl)18 145 y Fu(comm)o(ute)18 b(with)h Fo(h)p Fn(hf)390 152 y Fi(1)408 145 y Fn(;)8 b(:)g(:)g(:)d(;)j(hf)558 152 y Fk(n)580 145 y Fo(i)18 b Fu(=)g Fn(h)p Fo(h)p Fn(f)735 152 y Fi(1)754 145 y Fn(;)8 b(:)g(:)g(:)d(;)j(f)878 152 y Fk(n)900 145 y Fo(i)18 b Fu(and)h Fn(hf)1076 152 y Fi(1)1107 145 y Fu(+)13 b Fn(:)8 b(:)g(:)i Fu(+)i Fn(hf)1315 152 y Fk(n)1356 145 y Fu(=)19 b Fn(g)g Fu(=)f Fn(h)p Fu(\()p Fn(f)1570 152 y Fi(1)1601 145 y Fu(+)18 199 y Fn(:)8 b(:)g(:)g Fu(+)i Fn(f)148 206 y Fk(n)171 199 y Fu(\))p Fn(:)15 b Fu(The)g(second)h(part)e(follo)o(ws)i(from)e(the)h (comm)o(utativ)o(e)g(diagrams)533 301 y Fn(D)223 b(D)833 284 y Fk(n)p 585 286 196 2 v 739 285 a Fl(-)666 272 y Fm(e)-19 b Fk(\023)682 276 y Fj(i)1078 301 y Fn(D)p 869 286 V 869 285 a Fl(\033)942 276 y Fi(\001)969 264 y Fj(n)535 482 y Fn(A)227 b(A)830 466 y Fk(n)p 582 468 201 2 v 741 467 a Fl(-)666 454 y Fm(e)-19 b Fk(\023)682 458 y Fj(i)1080 482 y Fn(A)p 866 468 V 866 467 a Fl(\033)942 458 y Fi(\001)969 445 y Fj(n)p 551 435 2 119 v 552 435 a Fl(?)515 386 y Fk(k)p 824 435 V 825 435 a Fl(?)843 387 y Fk(k)862 374 y Fj(n)p 1096 435 V 1097 435 a Fl(?)1115 386 y Fk(k)630 565 y(f)646 569 y Fj(i)600 526 y Fl(Q)641 554 y(Q)683 582 y(Q)724 609 y(Q)735 617 y(Q)-42 b(s)807 664 y Fn(C)p 824 617 V 825 617 a Fl(?)843 538 y Fg(h)p Fk(f)872 542 y Fb(1)889 538 y Fk(;:::)o(;f)954 542 y Fj(n)974 538 y Fg(i)989 565 y Fk(f)1005 569 y Fb(1)1022 565 y Fi(+)p Fk(:::)p Fi(+)p Fk(f)1118 569 y Fj(n)1008 526 y Fl(\021)966 554 y(\021)925 582 y(\021)883 609 y(\021)872 617 y(\021)g(+)18 730 y Fu(b)o(y)15 b Fo(h)p Fn(f)121 737 y Fi(1)140 730 y Fn(;)8 b(:)g(:)g(:)t(;)g(f)263 737 y Fk(n)285 730 y Fo(i)p Fn(k)328 713 y Fk(n)363 730 y Fu(=)13 b Fo(h)p Fn(f)451 737 y Fi(1)470 730 y Fn(k)q(;)8 b(:)g(:)g(:)d(;)j(f)619 737 y Fk(n)641 730 y Fn(k)q Fo(i)15 b Fu(and)g(\()p Fn(f)827 737 y Fi(1)856 730 y Fu(+)10 b Fn(:)e(:)g(:)g Fu(+)j Fn(f)1032 737 y Fk(n)1055 730 y Fu(\))p Fn(k)i Fu(=)g Fn(f)1180 737 y Fi(1)1199 730 y Fn(k)e Fu(+)g Fn(:)d(:)g(:)g Fu(+)i Fn(f)1410 737 y Fk(n)1433 730 y Fn(k)q Fu(.)p 1613 703 24 2 v 1613 728 2 25 v 1635 728 V 1613 730 24 2 v 18 883 a Fv(Lemma)17 b(3.3)h Fe(Sums)e(of)h(summable)f(morphisms)g (satisfy)g(the)h(asso)n(ciative)e(law.)21 b(In)15 b(p)n(artic-)18 937 y(ular)20 b(if)f Fu(\()p Fn(f)203 944 y Fi(1)222 937 y Fn(;)8 b(f)265 944 y Fi(2)283 937 y Fn(;)g(f)326 944 y Fi(3)344 937 y Fu(\))18 b Fo(2)h(C)s Fu(\()p Fn(A;)8 b(B)r Fu(\))584 921 y Fi(3)620 937 y Fe(is)19 b(summable)g(then)g Fu(\()p Fn(f)1029 944 y Fi(1)1060 937 y Fu(+)13 b Fn(f)1130 944 y Fi(2)1149 937 y Fu(\))f(+)h Fn(f)1249 944 y Fi(3)1286 937 y Fu(=)19 b Fn(f)1362 944 y Fi(1)1393 937 y Fu(+)13 b Fn(f)1463 944 y Fi(2)1494 937 y Fu(+)g Fn(f)1564 944 y Fi(3)1601 937 y Fu(=)18 991 y Fn(f)40 998 y Fi(1)69 991 y Fu(+)d(\()p Fn(f)154 998 y Fi(2)183 991 y Fu(+)g Fn(f)250 998 y Fi(3)269 991 y Fu(\))p Fn(:)76 1074 y Fd(Pro)q(of:)57 b Fu(W)l(e)19 b(pro)o(v)o(e)g(only)h(the)g(second)g (statemen)o(t.)32 b(The)20 b(\014rst)f(follo)o(ws)h(b)o(y)f(standard)18 1128 y(reasoning.)h(Consider)c(the)f(follo)o(wing)h(comm)o(utativ)o(e)e (diagram)129 1425 y Fn(A)830 b(A)10 b Fo(\002)g Fn(A)p 176 1412 803 2 v 937 1411 a Fl(-)560 1394 y Fm(e)-20 b Fk(\023)575 1381 y Fb(2)575 1402 y(1)1492 1425 y Fn(A)p 1130 1412 349 2 v 1130 1411 a Fl(\033)1286 1394 y Fm(e)h Fk(\023)1302 1381 y Fb(2)1302 1402 y(2)652 1652 y Fn(A)10 b Fo(\002)h Fn(A)172 b(A)10 b Fo(\002)g Fn(A)h Fo(\002)f Fn(A)p 789 1639 146 2 v 893 1638 a Fl(-)832 1617 y Fm(e)-19 b Fk(\023)848 1605 y Fb(3)848 1625 y(1)p Fj(;)p Fb(1)1037 1199 y Fn(A)p 1054 1379 2 164 v 1054 1379 a Fl(?)1073 1307 y Fi(\001)489 1530 y(\001)228 1457 y Fl(P)269 1471 y(P)311 1484 y(P)352 1498 y(P)394 1512 y(P)435 1526 y(P)477 1540 y(P)518 1554 y(P)560 1567 y(P)601 1581 y(P)643 1595 y(P)650 1597 y(P)-42 b(q)p 1054 1606 V 1054 1606 a(?)955 1533 y Fi(\001)p Fg(\002)p Fi(id)1220 1536 y Fm(e)-19 b Fk(\023)1236 1523 y Fb(3)1236 1544 y(3)1404 1464 y Fl(\010)1362 1484 y(\010)1321 1505 y(\010)1279 1526 y(\010)1238 1547 y(\010)1196 1567 y(\010)1155 1588 y(\010)1118 1606 y(\010)-42 b(\031)913 1759 y Fg(h)p Fk(f)942 1763 y Fb(1)959 1759 y Fk(;f)985 1763 y Fb(2)1001 1759 y Fg(i)761 1698 y Fl(Q)803 1725 y(Q)844 1753 y(Q)886 1781 y(Q)927 1808 y(Q)965 1834 y(Q)g(s)481 1646 y Fk(f)497 1650 y Fb(1)513 1646 y Fi(+)p Fk(f)554 1650 y Fb(2)209 1464 y Fl(H)251 1484 y(H)292 1505 y(H)334 1526 y(H)376 1547 y(H)417 1567 y(H)459 1588 y(H)500 1609 y(H)542 1630 y(H)583 1650 y(H)625 1671 y(H)666 1692 y(H)708 1713 y(H)749 1733 y(H)791 1754 y(H)832 1775 y(H)874 1796 y(H)915 1816 y(H)949 1834 y(H)g(j)1310 1646 y Fk(f)1326 1650 y Fb(3)1435 1484 y Fl(\000)1394 1526 y(\000)1352 1567 y(\000)1311 1609 y(\000)1269 1650 y(\000)1228 1692 y(\000)1186 1733 y(\000)1145 1775 y(\000)1103 1816 y(\000)1086 1834 y(\000)g(\011)1036 1881 y Fn(B)p 1054 1834 V 1054 1834 a Fl(?)1073 1714 y Fg(h)p Fk(f)1102 1718 y Fb(1)1118 1714 y Fk(;f)1144 1718 y Fb(2)1161 1714 y Fk(;f)1187 1718 y Fb(3)1203 1714 y Fg(i)18 1947 y Fu(where)13 b(the)g(comp)q(osite)g(v)o(ertical)h(morphism)f(denotes)g(\()p Fn(f)1006 1954 y Fi(1)1030 1947 y Fu(+)5 b Fn(f)1092 1954 y Fi(2)1112 1947 y Fu(\))g(+)g Fn(f)1197 1954 y Fi(3)1229 1947 y Fu(\(and)13 b(also)g Fn(f)1444 1954 y Fi(1)1468 1947 y Fu(+)5 b Fn(f)1530 1954 y Fi(2)1555 1947 y Fu(+)g Fn(f)1617 1954 y Fi(3)18 2001 y Fu(in)16 b(case)f(the)h(diagram)f(comm)o(utes\).)k(The)d(morphisms)996 1999 y Fm(e)997 2001 y Fn(\023)1013 1984 y Fi(2)1047 2001 y Fu(are)f(morphisms)g(in)o(to)g Fn(A)1479 1984 y Fi(2)1514 2001 y Fu(and)1601 1999 y Fm(e)1602 2001 y Fn(\023)1618 1984 y Fi(3)18 2055 y Fu(in)o(to)d Fn(A)141 2038 y Fi(3)159 2055 y Fu(.)19 b(The)12 b(only)g(comm)o(utativit)o(y)g (whic)o(h)g(is)h(not)e(immediately)j(clear)e(is)g Fo(h)p Fn(f)1369 2062 y Fi(1)1388 2055 y Fn(;)c(f)1431 2062 y Fi(2)1449 2055 y Fn(;)g(f)1492 2062 y Fi(3)1510 2055 y Fo(i)1527 2053 y Fm(e)1528 2055 y Fn(\023)1544 2038 y Fi(3)1544 2066 y(1)p Fk(;)p Fi(1)1601 2055 y Fu(=)18 2109 y Fo(h)p Fn(f)58 2116 y Fi(1)76 2109 y Fn(;)g(f)119 2116 y Fi(2)137 2109 y Fo(i)p Fu(,)15 b(but)h(w)o(e)f(ha)o(v)o(e)g Fo(h)p Fn(f)479 2116 y Fi(1)497 2109 y Fn(;)8 b(f)540 2116 y Fi(2)558 2109 y Fn(;)g(f)601 2116 y Fi(3)619 2109 y Fo(i)636 2107 y Fm(e)637 2109 y Fn(\023)653 2092 y Fi(3)653 2120 y(1)p Fk(;)p Fi(1)697 2107 y Fm(e)698 2109 y Fn(\023)714 2092 y Fi(2)714 2120 y(1)746 2109 y Fu(=)13 b Fo(h)p Fn(f)834 2116 y Fi(1)852 2109 y Fn(;)8 b(f)895 2116 y Fi(2)914 2109 y Fn(;)g(f)957 2116 y Fi(3)975 2109 y Fo(i)992 2107 y Fm(e)993 2109 y Fn(\023)1009 2092 y Fi(3)1009 2120 y(1)1040 2109 y Fu(=)13 b Fn(f)1110 2116 y Fi(1)1142 2109 y Fu(=)g Fo(h)p Fn(f)1230 2116 y Fi(1)1249 2109 y Fn(;)8 b(f)1292 2116 y Fi(2)1310 2109 y Fo(i)1327 2107 y Fm(e)1328 2109 y Fn(\023)1344 2092 y Fi(2)1344 2120 y(1)1377 2109 y Fu(and)16 b(similarly)18 2163 y Fo(h)p Fn(f)58 2170 y Fi(1)76 2163 y Fn(;)8 b(f)119 2170 y Fi(2)137 2163 y Fn(;)g(f)180 2170 y Fi(3)199 2163 y Fo(i)216 2161 y Fm(e)217 2163 y Fn(\023)233 2146 y Fi(3)233 2174 y(1)p Fk(;)p Fi(1)276 2161 y Fm(e)277 2163 y Fn(\023)293 2146 y Fi(2)293 2174 y(2)325 2163 y Fu(=)14 b Fo(h)p Fn(f)414 2170 y Fi(1)433 2163 y Fn(;)8 b(f)476 2170 y Fi(2)494 2163 y Fo(i)511 2161 y Fm(e)512 2163 y Fn(\023)528 2146 y Fi(2)528 2174 y(2)546 2163 y Fu(.)21 b(By)16 b(the)g(uniqueness) h(of)f(the)f(factorization)h Fo(h)p Fn(f)1396 2170 y Fi(1)1414 2163 y Fn(;)8 b(f)1457 2170 y Fi(2)1475 2163 y Fo(i)16 b Fu(w)o(e)f(get)18 2217 y(the)g(required)h(comm)o(utativit)o (y)l(.)p 1613 2190 24 2 v 1613 2215 2 25 v 1635 2215 V 1613 2217 24 2 v 18 2370 a Fv(Lemma)h(3.4)h Fe(F)m(or)11 b(e)n(ach)h(morphism)g Fn(f)18 b Fo(2)13 b(C)s Fu(\()p Fn(A;)8 b(B)r Fu(\))i Fe(the)h(morphisms)h Fu(0)g Fe(and)f Fn(f)17 b Fe(ar)n(e)12 b(summable)18 2424 y(and)k(we)g(have)h Fu(0)10 b(+)g Fn(f)18 b Fu(=)13 b Fn(f)18 b Fu(=)13 b Fn(f)i Fu(+)c(0)p Fe(.)611 2738 y Ff(krs.tex)23 b(-)g(Date:)47 b(June)24 b(23,)f(1994)g(Time:)47 b(10:45)p eop %%Page: 11 11 11 10 bop 644 54 a Fh(Krull-Sc)o(hmid)o(t)9 b(Theorem)578 b Fu(11)76 145 y Fd(Pro)q(of:)35 b Fu(It)12 b(is)f(su\016cien)o(t)h(to) f(pro)o(v)o(e)f(0)r(+)r(id)k(=)f(id)f(for)f(then)g(0)r(+)r Fn(f)18 b Fu(=)13 b(0)p Fn(f)7 b Fu(+)r(id)i Fn(f)18 b Fu(=)13 b(\(0)r(+)r(id)q(\))p Fn(f)k Fu(=)18 199 y(id)8 b Fn(f)18 b Fu(=)13 b Fn(f)20 b Fu(b)o(y)15 b(Lemma)h(3.2.)j(But)c(the) g(factorization)g Fo(h)p Fu(0)p Fn(;)8 b Fu(id)p Fo(i)15 b Fu(is)g Fn(p)1142 206 y Fi(2)1173 199 y Fu(:)e Fn(A)d Fo(\002)g Fn(A)j Fo(\000)-7 b(!)12 b Fn(A)j Fu(since)583 319 y Fn(A)148 b(A)11 b Fo(\002)f Fn(A)p 631 306 122 2 v 711 305 a Fl(-)673 292 y Fm(e)-19 b Fk(\023)689 296 y Fb(1)1037 319 y Fn(A)p 903 306 V 903 305 a Fl(\033)945 292 y Fm(e)g Fk(\023)961 296 y Fb(2)810 548 y Fn(A)669 428 y Fi(0)632 379 y Fl(@)673 420 y(@)715 462 y(@)754 501 y(@)-42 b(R)p 826 501 2 164 v 31 w(?)845 422 y Fk(p)862 426 y Fb(2)969 429 y Fi(id)981 379 y Fl(\000)940 420 y(\000)898 462 y(\000)859 501 y(\000)g(\011)18 634 y Fu(comm)o(utes)14 b(b)o(y)i(de\014nition)h(of)547 632 y Fm(e)548 634 y Fn(\023)564 641 y Fi(1)598 634 y Fu(and)686 632 y Fm(e)687 634 y Fn(\023)703 641 y Fi(2)721 634 y Fu(.)j(Hence)c(0)10 b(+)h(id)i(=)g Fo(h)p Fu(0)p Fn(;)8 b Fu(id)p Fo(i)p Fu(\001)k(=)h Fn(p)1304 641 y Fi(2)1322 634 y Fu(\001)g(=)g(id)q(.)p 1613 608 24 2 v 1613 632 2 25 v 1635 632 V 1613 634 24 2 v 18 788 a Fv(Prop)q(osition)18 b(3.5)g Fe(L)n(et)e Fn(U)h Fu(=)c Fn(A)594 795 y Fi(1)623 788 y Fo(\002)e Fn(:)d(:)g(:)g Fo(\002)i Fn(A)811 795 y Fk(n)834 788 y Fe(.)21 b(Then)15 b Fu(id)1025 795 y Fk(U)1066 788 y Fu(=)1113 786 y Fm(e)1114 788 y Fn(\023)1130 795 y Fi(1)1148 788 y Fn(p)1171 795 y Fi(1)1200 788 y Fu(+)10 b Fn(:)e(:)g(:)h Fu(+)1353 786 y Fm(e)1354 788 y Fn(\023)1370 795 y Fk(n)1393 788 y Fn(p)1416 795 y Fk(n)1438 788 y Fe(.)76 894 y Fd(Pro)q(of:)39 b Fu(T)l(o)15 b(b)q(egin)i(the)e(pro)q(of)g(b)o(y)g(induction)i(assume)e Fn(U)j Fu(=)13 b Fn(A)d Fo(\002)g Fn(B)r Fu(.)20 b(The)15 b(diagram)582 1012 y Fn(U)150 b(U)15 b Fo(\002)c Fn(U)p 631 999 119 2 v 708 998 a Fl(-)672 985 y Fm(e)-19 b Fk(\023)688 989 y Fb(1)1036 1012 y Fn(U)p 904 999 V 904 998 a Fl(\033)945 985 y Fm(e)g Fk(\023)961 989 y Fb(2)583 1239 y Fn(A)98 b(A)11 b Fo(\002)f Fn(B)k Fu(=)f Fn(U)p 630 1226 72 2 v 660 1225 a Fl(-)644 1212 y Fm(e)-19 b Fk(\023)660 1216 y Fj(A)1037 1239 y Fn(A)p 950 1226 74 2 v 950 1225 a Fl(\033)965 1212 y Fm(e)f Fk(\023)980 1216 y Fj(B)p 599 1193 2 164 v 600 1193 a Fl(?)541 1115 y Fk(p)558 1119 y Fj(A)p 1053 1193 V 1054 1193 a Fl(?)1072 1115 y Fk(p)1089 1119 y Fj(B)p 826 1193 V 827 1193 a Fl(?)845 1118 y Fk(p)862 1122 y Fj(A)886 1118 y Fg(\002)p Fk(p)929 1122 y Fj(B)18 1327 y Fu(comm)o(utes,)14 b(whence)h Fn(p)425 1334 y Fk(A)462 1327 y Fo(\002)9 b Fn(p)529 1334 y Fk(B)570 1327 y Fu(=)k Fo(h)635 1325 y Fm(e)636 1327 y Fn(\023)652 1334 y Fk(A)679 1327 y Fn(p)702 1334 y Fk(A)729 1327 y Fn(;)749 1325 y Fm(e)749 1327 y Fn(\023)765 1334 y Fk(B)794 1327 y Fn(p)817 1334 y Fk(B)845 1327 y Fo(i)i Fu(and)g(id)1004 1334 y Fk(U)1044 1327 y Fu(=)e(\()p Fn(p)1133 1334 y Fk(A)1169 1327 y Fo(\002)c Fn(p)1236 1334 y Fk(B)1265 1327 y Fu(\)\001)j(=)1380 1325 y Fm(e)1381 1327 y Fn(\023)1397 1334 y Fk(A)1424 1327 y Fn(p)1447 1334 y Fk(A)1483 1327 y Fu(+)1527 1325 y Fm(e)1528 1327 y Fn(\023)1544 1334 y Fk(B)1573 1327 y Fn(p)1596 1334 y Fk(B)1624 1327 y Fu(.)18 1381 y(T)l(o)18 b(indicate)i(the)f (induction)i(step)d(assume)h Fn(V)28 b Fu(=)19 b Fn(A)12 b Fo(\002)h Fn(B)h Fo(\002)f Fn(C)21 b Fu(=)e Fn(U)e Fo(\002)c Fn(C)s Fu(.)30 b(Then)19 b(id)1554 1388 y Fk(V)1601 1381 y Fu(=)17 1433 y Fm(e)18 1435 y Fn(\023)34 1442 y Fk(U)62 1435 y Fn(p)85 1442 y Fk(U)127 1435 y Fu(+)176 1433 y Fm(e)177 1435 y Fn(\023)193 1442 y Fk(C)221 1435 y Fn(p)244 1442 y Fk(C)295 1435 y Fu(=)353 1433 y Fm(e)354 1435 y Fn(\023)370 1442 y Fk(U)406 1435 y Fu(id)443 1442 y Fk(U)479 1435 y Fn(p)502 1442 y Fk(U)544 1435 y Fu(+)593 1433 y Fm(e)594 1435 y Fn(\023)610 1442 y Fk(C)638 1435 y Fn(p)661 1442 y Fk(C)712 1435 y Fu(=)770 1433 y Fm(e)771 1435 y Fn(\023)787 1442 y Fk(U)815 1435 y Fu(\()832 1433 y Fm(e)833 1435 y Fn(\023)849 1418 y Fk(U)849 1446 y(A)877 1435 y Fn(p)900 1418 y Fk(U)900 1446 y(A)942 1435 y Fu(+)991 1433 y Fm(e)992 1435 y Fn(\023)1008 1418 y Fk(U)1008 1446 y(B)1036 1435 y Fn(p)1059 1418 y Fk(U)1059 1446 y(B)1088 1435 y Fu(\))p Fn(p)1129 1442 y Fk(U)1171 1435 y Fu(+)1219 1433 y Fm(e)1220 1435 y Fn(\023)1236 1442 y Fk(C)1264 1435 y Fn(p)1287 1442 y Fk(C)1339 1435 y Fu(=)1397 1433 y Fm(e)1398 1435 y Fn(\023)1414 1442 y Fk(U)1441 1433 y Fm(e)1442 1435 y Fn(\023)1458 1418 y Fk(U)1458 1446 y(A)1486 1435 y Fn(p)1509 1418 y Fk(U)1509 1446 y(A)1536 1435 y Fn(p)1559 1442 y Fk(U)1601 1435 y Fu(+)17 1487 y Fm(e)18 1489 y Fn(\023)34 1496 y Fk(U)61 1487 y Fm(e)62 1489 y Fn(\023)78 1472 y Fk(U)78 1500 y(B)106 1489 y Fn(p)129 1472 y Fk(U)129 1500 y(B)158 1489 y Fn(p)181 1496 y Fk(U)219 1489 y Fu(+)263 1487 y Fm(e)264 1489 y Fn(\023)280 1496 y Fk(C)308 1489 y Fn(p)331 1496 y Fk(C)371 1489 y Fu(=)418 1487 y Fm(e)419 1489 y Fn(\023)435 1496 y Fk(A)463 1489 y Fn(p)486 1496 y Fk(A)523 1489 y Fu(+)567 1487 y Fm(e)568 1489 y Fn(\023)584 1496 y Fk(B)613 1489 y Fn(p)636 1496 y Fk(B)674 1489 y Fu(+)719 1487 y Fm(e)720 1489 y Fn(\023)736 1496 y Fk(C)764 1489 y Fn(p)787 1496 y Fk(C)815 1489 y Fu(.)p 1613 1462 24 2 v 1613 1487 2 25 v 1635 1487 V 1613 1489 24 2 v 18 1642 a Fv(Example)f(3.6)g Fe(We)e(want)g(to)h(pr)n(ove)f(in)g (the)g(c)n(ase)g Fo(C)f Fu(=)e Fa(Gr)p Fe(,)j(the)g(c)n(ate)n(gory)g (of)h(gr)n(oups,)f(that)18 1696 y(two)j(morphisms)g Fn(f)r(;)8 b(g)18 b Fu(:)e Fn(A)h Fo(\000)-7 b(!)16 b Fn(B)21 b Fe(ar)n(e)d(summable)h(if)f(and)h(only)f(if)h Fn(f)5 b Fu(\()p Fn(x)p Fu(\))p Fn(g)r Fu(\()p Fn(y)r Fu(\))14 b(=)k Fn(g)r Fu(\()p Fn(y)r Fu(\))p Fn(f)5 b Fu(\()p Fn(x)p Fu(\))18 1750 y Fe(for)25 b(al)r(l)g Fn(x;)8 b(y)29 b Fo(2)g Fn(A)p Fe(.)47 b(Given)25 b Fo(h)p Fn(f)r(;)8 b(g)r Fo(i)23 b Fe(we)i(have)g Fo(h)p Fn(f)r(;)8 b(g)r Fo(i)p Fu(\()p Fn(x;)g(y)r Fu(\))26 b(=)i Fo(h)p Fn(f)r(;)8 b(g)r Fo(i)p Fu(\(\()p Fn(x;)g(e)p Fu(\))14 b Fo(\001)i Fu(\()p Fn(e;)8 b(y)r Fu(\)\))26 b(=)18 1804 y Fo(h)p Fn(f)r(;)8 b(g)r Fo(i)p Fu(\()p Fn(x;)g(e)p Fu(\))k Fo(\001)k(h)p Fn(f)r(;)8 b(g)r Fo(i)p Fu(\()p Fn(e;)g(y)r Fu(\))23 b(=)j Fn(f)5 b Fu(\()p Fn(x)p Fu(\))15 b Fo(\001)g Fn(g)r Fu(\()p Fn(y)r Fu(\))p Fe(.)41 b(Sinc)n(e)23 b Fu(\()p Fn(x;)8 b(e)p Fu(\))22 b Fe(and)i Fu(\()p Fn(e;)8 b(y)r Fu(\))22 b Fe(c)n(ommute)i(we)g(get)18 1858 y Fn(f)5 b Fu(\()p Fn(x)p Fu(\))p Fn(g)r Fu(\()p Fn(y)r Fu(\))24 b(=)j Fn(g)r Fu(\()p Fn(y)r Fu(\))p Fn(f)5 b Fu(\()p Fn(x)p Fu(\))p Fe(.)41 b(Conversely)23 b(it)g(is)h(a)g(wel)r(l-known)f (exer)n(cise)g(that)h(this)g(c)n(ondi-)18 1912 y(tion)c(implies)g(that) i Fo(h)p Fn(f)r(;)8 b(g)r Fo(i)19 b Fe(is)h(a)h(homomorphism.)36 b(The)20 b(sum)h Fn(f)e Fu(+)13 b Fn(g)22 b Fe(is)f(then)f(de\014ne)n (d)g(by)18 1966 y Fu(\()p Fn(f)15 b Fu(+)10 b Fn(g)r Fu(\)\()p Fn(x)p Fu(\))h(=)i Fn(f)5 b Fu(\()p Fn(x)p Fu(\))p Fn(g)r Fu(\()p Fn(x)p Fu(\))p Fe(.)246 2096 y Fv(4)53 b(Idemp)q(oten)o(t)17 b(morphisms)f(and)i(the)g(Fitting)h (Lemma)18 2184 y Fu(First)d(w)o(e)f(need)i(some)f(facts)f(ab)q(out)h (coimages.)23 b(The)16 b(coimage)g(of)g Fn(f)j Fu(:)14 b Fn(A)g Fo(\000)-7 b(!)13 b Fn(B)18 b Fu(is)e(de\014ned)18 2238 y(as)i(the)h(di\013erence)i(cok)o(ernel)e(of)g(the)g(k)o(ernel)h (pair)f(of)g Fn(f)24 b Fu(\([2])17 b(p.70,)i(Lemma)g(4a\)\).)30 b(In)20 b(the)18 2309 y(canonical)d(factorization)e Fn(A)551 2282 y Fk(f)570 2269 y Fc(0)530 2309 y Fo(\000)-7 b(!)13 b Fu(Coim\()p Fn(f)5 b Fu(\))830 2284 y Fk(\023)799 2309 y Fo(\000)-7 b(!)13 b Fn(B)k Fu(of)f Fn(f)5 b Fu(,)15 b Fn(\023)h Fu(fails)h(to)e(b)q(e)h(a)g(monomorphism)18 2363 y(in)g(general.)611 2738 y Ff(krs.tex)23 b(-)g(Date:)47 b(June)24 b(23,)f(1994)g(Time:)47 b(10:45)p eop %%Page: 12 12 12 11 bop 18 54 a Fu(12)510 b Fh(Bo)q(do)11 b(P)o(areigis)f(and)g(Helm) o(ut)h(R\177)-18 b(ohrl)18 145 y Fv(Lemma)17 b(4.1)h Fe(Given)12 b Fn(f)18 b Fu(:)13 b Fn(A)f Fo(\000)-7 b(!)12 b Fn(B)j Fe(and)e Fn(g)g Fu(:)g Fn(B)h Fo(\000)-7 b(!)12 b Fn(C)s Fe(,)h(then)g(ther)n(e)g(is)f(a)h(unique)g(morphism)18 199 y Fn(k)h Fu(:)e(Coim\()p Fn(g)r(f)5 b Fu(\))11 b Fo(\000)-7 b(!)12 b Fu(Coim\()p Fn(g)r Fu(\))j Fe(such)i(that)588 305 y Fn(A)192 b(B)p 636 290 165 2 v 759 289 a Fl(-)708 275 y Fk(f)1042 305 y Fn(C)p 864 290 164 2 v 986 289 a Fl(-)937 275 y Fk(g)509 528 y Fu(Coim\()p Fn(g)r(f)5 b Fu(\))279 b(Coim\()p Fn(g)r Fu(\))p 715 517 249 2 v 922 516 a Fl(-)830 507 y Fk(k)p 604 485 2 164 v 605 485 a Fl(?)864 362 y(@)906 404 y(@)947 446 y(@)986 484 y(@)-42 b(R)p 1059 485 V 1059 362 a(6)669 485 y(\010)710 464 y(\010)752 443 y(\010)793 422 y(\010)835 401 y(\010)876 381 y(\010)918 360 y(\010)954 342 y(\010)g(*)18 615 y Fe(c)n(ommutes.)21 b(If)16 b Fn(f)21 b Fe(is)16 b(an)g(isomorphism)h (then)f(so)g(is)f Fn(k)q Fe(.)76 712 y Fd(Pro)q(of:)46 b Fu(This)17 b(is)h(an)f(easy)f(exercise)i(in)g(univ)o(ersal)g(prop)q (erties)g(of)e(di\013erence)i(cok)o(ernels)18 766 y(and)d(k)o(ernel)h (pairs.)p 1613 739 24 2 v 1613 764 2 25 v 1635 764 V 1613 766 24 2 v 76 919 a(One)g(pro)o(v)o(es)e(just)h(as)g(easily)18 1016 y Fv(Lemma)i(4.2)h Fe(Given)h Fn(g)h Fu(:)e Fn(A)g Fo(\000)-7 b(!)17 b Fn(B)22 b Fe(and)d Fn(f)k Fu(:)18 b Fn(B)i Fo(\000)-7 b(!)18 b Fn(C)s Fe(,)i(ther)n(e)f(is)g(a)h(unique)f (morphism)18 1070 y Fn(k)14 b Fu(:)e(Coim\()p Fn(g)r Fu(\))f Fo(\000)-7 b(!)12 b Fu(Coim\()p Fn(f)5 b(g)r Fu(\))15 b Fe(such)i(that)574 1176 y Fn(A)192 b(B)p 622 1161 165 2 v 745 1160 a Fl(-)696 1146 y Fk(g)1028 1176 y Fn(C)p 850 1161 164 2 v 972 1160 a Fl(-)923 1146 y Fk(f)509 1399 y Fu(Coim\()p Fn(g)r Fu(\))279 b(Coim\()p Fn(f)5 b(g)r Fu(\))p 687 1388 249 2 v 894 1387 a Fl(-)803 1378 y Fk(k)p 591 1355 2 164 v 592 1355 a Fl(?)-11 b(\000)665 1314 y(\000)706 1272 y(\000)745 1233 y(\000)-42 b(\022)p 1045 1355 V 259 w(6)655 1213 y(H)697 1233 y(H)738 1254 y(H)780 1275 y(H)821 1296 y(H)863 1316 y(H)904 1337 y(H)941 1355 y(H)g(j)18 1486 y Fe(c)n(ommutes.)21 b(If)16 b Fn(f)21 b Fe(is)16 b(an)g(isomorphism)h(then)f(so)g(is)f Fn(k)q Fe(.)18 1583 y Fv(De\014nition)k(4.3)f Fu(Let)c Fn(f)k Fu(:)12 b Fn(A)h Fo(\000)-7 b(!)12 b Fn(A)i Fu(b)q(e)h(idemp)q(oten)o (t.)21 b(W)l(e)14 b(sa)o(y)g(that)f Fn(f)20 b Fu(satis\014es)14 b(condition)18 1637 y(\()p Fn(G)72 1644 y Fk(f)93 1637 y Fu(\))k(if)h(the)g(canonical)h(morphism)e(Ke)q(\()p Fn(f)5 b Fu(\))801 1604 y Fm(`)847 1637 y Fu(Coim\()p Fn(f)g Fu(\))18 b Fo(\000)-7 b(!)17 b Fn(A)i Fu(is)g(a)f(di\013erence)i (cok)o(ernel)18 1691 y(and)15 b(Coim\()p Fn(f)5 b Fu(\))12 b Fo(\000)-7 b(!)12 b Fn(A)k Fu(is)f(a)g(k)o(ernel.)76 1788 y(In)23 b(the)f(category)g(of)f(groups)h Fv(Gr)g Fu(the)h(\014rst)f(condition)h(is)g(alw)o(a)o(ys)f(satis\014ed,)i (indeed)18 1842 y Fn(a)14 b Fo(7!)g Fn(af)5 b Fu(\()p Fn(a)209 1825 y Fg(\000)p Fi(1)253 1842 y Fu(\))11 b Fo(\003)f Fn(f)5 b Fu(\()p Fn(a)p Fu(\))14 b Fo(2)h Fu(Ke\()p Fn(f)5 b Fu(\))587 1809 y Fm(`)633 1842 y Fu(Coim\()p Fn(f)g Fu(\))16 b(is)g(a)g(section)h(for)e(the)i(giv)o(en)f(morphism.) 23 b(The)18 1895 y(second)e(condition)g(is)g(in)g Fv(Gr)f Fu(equiv)m(alen)o(t)i(to)e Fn(f)25 b Fu(b)q(eing)d(normal)e(\(see)g (4.7\).)34 b(In)21 b Fv(Mo)p Fu(,)g(the)18 1949 y(category)13 b(of)g(monoids,)h(let)h Fn(M)j Fu(=)13 b(\()p Fn(Z)q(=)p Fu(3)p Fn(Z)s Fu(\))764 1933 y Fg(\002)804 1949 y Fu(b)q(e)i(the)f(m)o (ultiplicativ)o(e)i(monoid)f(of)e Fn(Z)q(=)p Fu(3)p Fn(Z)k Fu(and)18 2003 y Fn(f)j Fu(:)14 b Fn(M)20 b Fo(\000)-8 b(!)14 b Fn(M)22 b Fu(b)q(e)17 b(giv)o(en)g(b)o(y)f Fn(f)5 b Fu(\()p 596 1969 23 2 v(0\))14 b(=)p 701 1969 V 15 w(0,)i Fn(f)5 b Fu(\()p 798 1969 V(1\))14 b(=)h Fn(f)5 b Fu(\()p 948 1969 V(2\))14 b(=)p 1053 1969 V 15 w(1.)23 b(Then)17 b Fn(f)k Fu(is)c(idemp)q(oten)o(t)h(and)18 2057 y(Ke\()p Fn(f)5 b Fu(\))144 2025 y Fm(`)190 2057 y Fu(Coim\()p Fn(f)g Fu(\))17 b Fo(\000)-8 b(!)17 b Fn(M)23 b Fu(is)18 b(surjectiv)o(e.)28 b(But)18 b(Coim\()p Fn(f)5 b Fu(\))16 b Fo(\000)-7 b(!)16 b Fn(M)23 b Fu(fails)18 b(to)f(b)q(e)i(a)e(k)o(ernel.)18 2111 y(T)l(o)f(see)g(this)g(observ)o (e)g(that)f(Coim\()p Fn(f)5 b Fu(\))14 b(=)g Fo(f)p 766 2077 V Fu(0)p Fn(;)p 810 2077 V 8 w Fu(1)o Fo(g)h Fu(and)i(the)f(k)o (ok)o(ernel)g(of)f(Coim)q(\()p Fn(f)5 b Fu(\))13 b Fo(\000)-7 b(!)13 b Fn(M)21 b Fu(is)18 2165 y Fn(M)e Fo(\000)-7 b(!)14 b(f)p 192 2131 V Fu(1)o Fo(g)p Fu(.)23 b(The)17 b(k)o(ernel)g(of)f(this)g(morphism)h(is)g(id)e(:)f Fn(M)19 b Fo(\000)-7 b(!)14 b Fn(M)5 b Fu(,)16 b(but)h(not)f(Coim\()p Fn(f)5 b Fu(\))14 b Fo(\000)-8 b(!)18 2219 y Fn(M)5 b Fu(.)76 2273 y(No)o(w)14 b(let)i Fo(C)i Fu(b)q(e)e(again)f(as)g(in)h (section)g(1.)18 2370 y Fv(Lemma)h(4.4)h Fe(L)n(et)k Fn(f)29 b Fu(:)24 b Fn(A)g Fo(\000)-7 b(!)24 b Fn(A)e Fe(b)n(e)h(idemp)n(otent)f(with)h Fu(\()p Fn(G)1133 2377 y Fk(f)1154 2370 y Fu(\))p Fe(.)40 b(Then)22 b Fn(A)i Fu(=)h(Ke\()p Fn(f)5 b Fu(\))14 b Fo(\002)18 2424 y Fu(Coim\()p Fn(f)5 b Fu(\))p Fn(:)611 2738 y Ff(krs.tex)23 b(-)g(Date:)47 b(June)24 b(23,)f(1994)g(Time:)47 b(10:45)p eop %%Page: 13 13 13 12 bop 644 54 a Fh(Krull-Sc)o(hmid)o(t)9 b(Theorem)578 b Fu(13)76 145 y Fd(Pro)q(of:)36 b Fu(Ke\()p Fn(f)5 b Fu(\))12 b Fo(\000)-7 b(!)12 b Fn(A)f Fu(and)h(Coim\()p Fn(f)5 b Fu(\))12 b Fo(\000)-7 b(!)12 b Fn(A)f Fu(are)h(k)o(ernels.)19 b(F)l(urthermore)11 b(\()p Fn(G)1448 152 y Fk(f)1469 145 y Fu(\))g(implies)18 199 y Fn(A)i Fu(=)g(Ke\()p Fn(f)5 b Fu(\))10 b Fo([)g Fu(Coim\()p Fn(f)5 b Fu(\).)19 b(Next)d(w)o(e)f (sho)o(w)f(Ke)q(\()p Fn(f)5 b Fu(\))k Fo(\\)i Fu(Coim\()p Fn(f)5 b Fu(\))12 b(=)h(0.)19 b(In)d(the)g(diagram)573 319 y Fn(A)130 b Fu(Coim\()p Fn(f)5 b Fu(\))p 620 308 100 2 v 678 307 a Fl(-)655 293 y Fk(f)674 281 y Fc(0)1027 319 y Fn(A)p 915 308 V 973 307 a Fl(-)958 299 y Fk(\023)890 427 y(\023)849 381 y Fl(@)890 422 y(@)932 464 y(@)971 503 y(@)-42 b(R)1027 550 y Fn(A)p 1043 503 2 164 v 1044 503 a Fl(?)1062 428 y Fk(f)18 643 y Fu(w)o(e)18 b(ha)o(v)o(e)g Fn(\023f)239 627 y Fg(0)269 643 y Fu(=)g Fn(f)23 b Fu(=)18 b Fn(f)447 627 y Fi(2)484 643 y Fu(=)g Fn(f)5 b(\023f)607 627 y Fg(0)637 643 y Fu(hence)20 b Fn(\023)e Fu(=)g Fn(f)5 b(\023)p Fu(,)19 b(since)h Fn(f)1071 627 y Fg(0)1101 643 y Fu(is)f(an)f(epimorphism.)31 b(In)19 b(the)18 697 y(comm)o(utativ)o(e)14 b(diagram)439 799 y(Ke)q(\()p Fn(f)5 b Fu(\))k Fo(\\)i Fu(Coim\()p Fn(f)5 b Fu(\))139 b(Ke)q(\()p Fn(f)5 b Fu(\))p 790 789 110 2 v 858 788 a Fl(-)524 1026 y Fu(Coim\()p Fn(f)g Fu(\))261 b Fn(A)p 706 1016 236 2 v 900 1015 a Fl(-)817 1006 y Fk(\023)1181 1030 y Fn(A)p 1002 1016 167 2 v 1127 1015 a Fl(-)1075 1000 y Fk(f)p 607 983 2 164 v 608 983 a Fl(?)626 908 y Fk(j)p 970 983 V 971 983 a Fl(?)1113 911 y Fi(0)1003 861 y Fl(@)1045 903 y(@)1086 944 y(@)1125 983 y(@)-42 b(R)18 1131 y Fu(w)o(e)13 b(ha)o(v)o(e)h Fn(\023j)h Fu(=)e Fn(f)5 b(\023j)15 b Fu(=)e(0)h(hence)h(Ke\()p Fn(f)5 b Fu(\))i Fo(\\)g Fu(Coim)q(\()p Fn(f)e Fu(\))12 b(=)h(0.)19 b(Hence)c(Theorem)e(2.7)g(\014nishes)j(the)18 1184 y(pro)q(of.)p 1613 1158 24 2 v 1613 1183 2 25 v 1635 1183 V 1613 1185 24 2 v 76 1352 a(Let)21 b Fn(f)28 b Fu(:)22 b Fn(A)h Fo(\000)-8 b(!)22 b Fn(A)f Fu(b)q(e)h(an)f(endomorphism.)39 b(Then)21 b Fn(f)26 b Fu(has)21 b(a)g(factorization)g Fn(A)1585 1325 y Fk(f)1604 1313 y Fc(0)1564 1352 y Fo(\000)-8 b(!)18 1406 y Fu(Coim\()p Fn(f)5 b Fu(\))263 1381 y Fk(\023)233 1406 y Fo(\000)-7 b(!)45 b Fn(A)p Fu(.)81 b(By)36 b(Lemma)f(4.1)g (there)g(is)h(also)g(a)f(canonical)h(morphism)18 1460 y Fn(\023)34 1467 y Fk(n)71 1460 y Fu(:)15 b(Coim\()p Fn(f)250 1444 y Fk(n)272 1460 y Fu(\))g Fo(\000)-8 b(!)15 b Fu(Coim\()p Fn(f)544 1444 y Fk(n)p Fg(\000)p Fi(1)609 1460 y Fu(\))h(and)g(b)o(y)h(Lemma)f(4.2)g(there)g(is)h(a)g(canonical)g (morphism)18 1514 y Fn(f)45 1498 y Fg(0)40 1525 y Fk(n)75 1514 y Fu(:)c(Coim\()p Fn(f)252 1498 y Fk(n)p Fg(\000)p Fi(1)317 1514 y Fu(\))f Fo(\000)-7 b(!)12 b Fu(Coim\()p Fn(f)584 1498 y Fk(n)606 1514 y Fu(\).)18 1630 y Fv(De\014nition)19 b(4.5)f Fu(An)g(endomorphism)g Fn(f)j Fu(:)16 b Fn(A)h Fo(\000)-8 b(!)16 b Fn(A)i Fu(is)g(called)h Fe(b)n(ounde)n(d)p Fu(,)e(if)h(the)f(families)18 1684 y(\()p Fn(\023)52 1691 y Fk(n)87 1684 y Fu(:)12 b(Coim\()p Fn(f)263 1667 y Fk(n)286 1684 y Fu(\))g Fo(\000)-7 b(!)12 b Fu(Coim\()p Fn(f)553 1667 y Fk(n)p Fg(\000)p Fi(1)618 1684 y Fu(\)\))f(and)h(\()p Fn(f)795 1667 y Fg(0)790 1695 y Fk(n)825 1684 y Fu(:)h(Coim\()p Fn(f)1002 1667 y Fk(n)p Fg(\000)p Fi(1)1067 1684 y Fu(\))f Fo(\000)-7 b(!)12 b Fu(Coim\()p Fn(f)1334 1667 y Fk(n)1356 1684 y Fu(\)\))g(b)q(ecome)g(sta-)18 1738 y(tionary)h(\(i.e.)19 b(there)14 b(is)g Fn(n)455 1745 y Fi(0)487 1738 y Fu(suc)o(h)g(that)f (for)g(all)h Fn(n)f Fo(\025)g Fn(n)929 1745 y Fi(0)961 1738 y Fu(b)q(oth)h Fn(\023)1083 1745 y Fk(n)1119 1738 y Fu(and)f Fn(f)1232 1721 y Fg(0)1227 1749 y Fk(n)1264 1738 y Fu(are)g(isomorphisms\),)18 1792 y(and)h(if)h(for)f(eac)o(h)g Fn(n)h Fu(there)g(is)f Fn(r)g Fo(\025)f Fn(n)h Fu(suc)o(h)h(that)f (Ke\()p Fn(f)942 1775 y Fk(r)961 1792 y Fu(\))987 1760 y Fm(`)1033 1792 y Fu(Coim\()p Fn(f)1184 1775 y Fk(r)1202 1792 y Fu(\))e Fo(\000)-7 b(!)12 b Fn(A)j Fu(is)f(a)h(di\013erence)18 1846 y(cok)o(ernel)h(and)f(Coim\()p Fn(f)434 1829 y Fk(r)452 1846 y Fu(\))e Fo(\000)-8 b(!)13 b Fn(A)i Fu(is)g(a)g(k)o(ernel.)18 1962 y Fv(Prop)q(osition)j(4.6)g Fu(\(Fitting)f(Lemma\))h Fe(L)n(et)f Fn(f)k Fu(:)16 b Fn(A)g Fo(\000)-8 b(!)16 b Fn(A)i Fe(b)n(e)f(b)n(ounde)n(d.)26 b(Then)17 b(for)i(every)18 2016 y Fn(n)45 2023 y Fi(0)76 2016 y Fo(2)13 b Fn(N)21 b Fe(ther)n(e)16 b(is)g(an)g Fn(n)d Fo(\025)g Fn(n)519 2023 y Fi(0)554 2016 y Fe(such)k(that)580 2124 y Fn(A)c Fu(=)g(Ke\()p Fn(f)775 2105 y Fk(n)798 2124 y Fu(\))c Fo(\002)i Fu(Coim\()p Fn(f)1022 2105 y Fk(n)1044 2124 y Fu(\))p Fn(:)76 2240 y Fd(Pro)q(of:)36 b Fu(The)12 b(c)o(hains)h(\()p Fn(\023)494 2247 y Fk(n)517 2240 y Fu(\))e(and)h(\()p Fn(f)676 2223 y Fg(0)671 2251 y Fk(n)694 2240 y Fu(\))g(b)q(ecome)g(stationary)f(for)h(all)h Fn(n)g Fo(\025)g Fn(n)1338 2247 y Fi(0)1357 2240 y Fu(.)18 b(In)13 b(particular)18 2294 y(w)o(e)i(ha)o(v)o(e)g(an)g(in)o(v)o(erse)g(\()p Fn(f)448 2277 y Fg(0)443 2305 y Fk(n)466 2294 y Fn(\023)482 2301 y Fk(n)505 2294 y Fu(\))523 2277 y Fg(\000)p Fk(n)586 2294 y Fu(of)g(\()p Fn(f)683 2277 y Fg(0)678 2305 y Fk(n)700 2294 y Fn(\023)716 2301 y Fk(n)739 2294 y Fu(\))757 2277 y Fk(n)779 2294 y Fu(.)20 b(The)15 b(morphism)388 2424 y Fn(')e Fu(:)f Fn(A)g Fo(\000)-7 b(!)12 b Fu(Coim\()p Fn(f)739 2405 y Fk(n)762 2424 y Fu(\))792 2395 y Fi(\()p Fk(f)824 2382 y Fc(0)821 2403 y Fj(n)842 2395 y Fk(\023)854 2399 y Fj(n)874 2395 y Fi(\))887 2382 y Fc(\000)p Fj(n)824 2424 y Fo(\000)-7 b(!)45 b Fu(Coim\()p Fn(f)1094 2405 y Fk(n)1116 2424 y Fu(\))12 b Fo(\000)-7 b(!)12 b Fn(A)611 2738 y Ff(krs.tex)23 b(-)g(Date:)47 b(June)24 b(23,)f(1994)g(Time:)47 b(10:45)p eop %%Page: 14 14 14 13 bop 18 54 a Fu(14)510 b Fh(Bo)q(do)11 b(P)o(areigis)f(and)g(Helm) o(ut)h(R\177)-18 b(ohrl)18 145 y Fu(is)16 b(idemp)q(oten)o(t)f(whic)o (h)h(follo)o(ws)g(from)e(the)h(comm)o(utativ)o(e)g(diagram)354 256 y Fn(A)420 b(A)p 402 241 394 2 v 402 240 a Fl(\033)578 226 y Fk(f)597 214 y Fj(n)276 479 y Fu(Coim\()p Fn(f)427 462 y Fk(n)449 479 y Fu(\))267 b(Coim\()p Fn(f)885 462 y Fk(n)907 479 y Fu(\))p 480 468 237 2 v 675 467 a Fl(-)529 450 y Fi(\()p Fk(f)561 438 y Fc(0)558 459 y Fj(n)579 450 y Fk(\023)591 454 y Fj(n)612 450 y Fi(\))625 438 y Fc(\000)p Fj(n)354 710 y Fn(A)420 b(A)p 402 696 394 2 v 402 695 a Fl(\033)578 680 y Fk(f)597 668 y Fj(n)1263 710 y Fn(A)p 856 696 V 856 695 a Fl(\033)942 725 y Fk(f)961 712 y Fj(n)730 933 y Fu(Coim\()p Fn(f)881 917 y Fk(n)903 933 y Fu(\))267 b(Coim\()p Fn(f)1339 917 y Fk(n)1361 933 y Fu(\))p 935 923 237 2 v 1130 922 a Fl(-)984 957 y Fi(\()p Fk(f)1016 945 y Fc(0)1013 966 y Fj(n)1033 957 y Fk(\023)1045 961 y Fj(n)1066 957 y Fi(\))1079 945 y Fc(\000)p Fj(n)808 1165 y Fn(A)421 b(A)p 856 1150 394 2 v 856 1149 a Fl(\033)1033 1179 y Fk(f)1052 1167 y Fj(n)p 370 436 2 164 v 371 436 a Fl(?)p 370 663 V 371 663 a(?)p 825 436 V 825 436 a(?)p 825 663 V 825 663 a(?)p 825 890 V 825 890 a(?)p 825 1117 V 825 1117 a(?)p 1279 890 V 1280 890 a(?)p 1279 1117 V 1280 1117 a(?)1081 478 y Fi(id)857 314 y Fl(@)899 355 y(@)940 397 y(@)982 438 y(@)1023 480 y(@)1065 521 y(@)1106 563 y(@)1148 604 y(@)1189 646 y(@)1206 663 y(@)-42 b(R)857 541 y(@)899 582 y(@)940 624 y(@)982 665 y(@)1023 707 y(@)1065 748 y(@)1106 790 y(@)1148 832 y(@)1189 873 y(@)1206 890 y(@)g(R)967 591 y Fi(id)18 1271 y Fu(in)17 b(whic)o(h)g Fn(')233 1254 y Fi(2)266 1271 y Fu(:)e Fn(A)g Fo(\000)-8 b(!)15 b Fn(A)h Fu(is)h(easily)g(iden)o (ti\014ed.)26 b(The)17 b(diagonal)g(Coim\()p Fn(f)1303 1254 y Fk(n)1325 1271 y Fu(\))e Fo(\000)-8 b(!)15 b Fu(Coim\()p Fn(f)1597 1254 y Fk(n)1619 1271 y Fu(\))18 1325 y(is)g(the)f(iden)o (tit)o(y)h(and)f(giv)o(es)h(a)f(comm)o(utativ)o(e)f(lo)o(w)o(er)h (triangle)h(since)g Fn(A)e Fo(\000)-7 b(!)12 b Fu(Coim\()p Fn(f)1490 1308 y Fk(n)1512 1325 y Fu(\))i(is)h(an)18 1389 y(epimorphism.)29 b(Th)o(us)17 b Fn(A)g Fo(\000)-7 b(!)16 b Fu(Coim\()p Fn(f)723 1373 y Fk(n)746 1389 y Fu(\))g Fo(\000)-7 b(!)16 b Fu(Coim\()p Fn(f)1021 1373 y Fk(n)1044 1389 y Fu(\))1101 1364 y Fi(id)1078 1389 y Fo(\000)-7 b(!)16 b Fu(Coim\()p Fn(f)1319 1373 y Fk(n)1342 1389 y Fu(\))g Fo(\000)-7 b(!)16 b Fn(A)i Fu(in)h(the)18 1443 y(diagram)c(is)g Fn(')p Fu(.)76 1497 y(The)h(only)g(remaining)h (problem)f(is)g(the)g(comm)o(utativit)o(y)f(of)g(the)h(rectangles.)21 b(It)16 b(follo)o(ws)18 1551 y(from)e(the)i(diagram)263 1667 y Fn(A)330 b(A)p 311 1653 303 2 v 311 1652 a Fl(\033)424 1637 y Fk(f)443 1625 y Fj(n)p Fc(\000)p Fb(1)185 1845 y Fu(Coim\()p Fn(f)336 1828 y Fk(n)358 1845 y Fu(\))176 b(Coim\()p Fn(f)703 1828 y Fk(n)725 1845 y Fu(\))p 389 1834 146 2 v 389 1833 a Fl(\033)263 2031 y Fn(A)330 b(A)p 311 2016 303 2 v 311 2015 a Fl(\033)424 2001 y Fk(f)443 1988 y Fj(n)p Fc(\000)p Fb(1)1354 1667 y Fn(A)p 674 1653 666 2 v 674 1652 a Fl(\033)997 1637 y Fk(f)894 1846 y Fu(Coim\()p Fn(f)1045 1829 y Fk(n)p Fg(\000)p Fi(1)1110 1846 y Fu(\))p 753 1834 125 2 v 753 1833 a Fl(\033)796 1866 y Fk(f)815 1854 y Fc(0)812 1875 y Fj(n)1279 1846 y Fu(Coim\()p Fn(f)1430 1829 y Fk(n)1452 1846 y Fu(\))p 1138 1834 V 1138 1833 a Fl(\033)1183 1821 y Fk(\023)1195 1825 y Fj(n)1354 2031 y Fn(A)p 674 2016 666 2 v 674 2015 a Fl(\033)997 2001 y Fk(f)p 280 1802 2 119 v 280 1802 a Fl(?)p 280 1984 V 280 1984 a(?)p 643 1802 V 644 1802 a(?)p 643 1984 V 644 1984 a(?)p 1370 1802 V 1371 1802 a(?)p 1370 1984 V 1371 1984 a(?)707 1704 y(H)749 1725 y(H)790 1746 y(H)832 1767 y(H)873 1787 y(H)902 1802 y(H)-42 b(j)1071 1886 y(H)1112 1907 y(H)1154 1928 y(H)1195 1948 y(H)1237 1969 y(H)1266 1984 y(H)g(j)18 2116 y Fu(where)14 b(the)f(left)h(part)f(is)h(an)g(\()p Fn(n)7 b Fo(\000)g Fu(1\)-fold)13 b(rep)q(etition)i(of)e(the)h(righ)o(t)g(part)f(and)g (the)h(righ)o(t)f(part)18 2170 y(comm)o(utes)h(b)o(y)i(Lemma)f(4.1)f (and)h(Lemma)h(4.2)e(with)h Fn(g)f Fu(=)f Fn(f)1068 2154 y Fk(n)p Fg(\000)p Fi(1)1134 2170 y Fu(.)76 2224 y(F)l(or)e(suitably)j (large)e Fn(n)g Fu(w)o(e)g(ha)o(v)o(e)g(also)g(Coim\()p Fn(f)877 2208 y Fk(n)899 2224 y Fu(\))g Fo(\000)-7 b(!)12 b Fn(A)g Fu(a)g(k)o(ernel.)20 b(De\014ne)13 b Fn(')1415 2208 y Fg(0)1439 2224 y Fu(:=)f(\()p Fn(A)h Fo(\000)-8 b(!)18 2300 y Fu(Coim\()p Fn(f)169 2283 y Fk(n)191 2300 y Fu(\))231 2270 y Fi(\()p Fk(f)263 2257 y Fc(0)260 2278 y Fj(n)281 2270 y Fk(\023)293 2274 y Fj(n)314 2270 y Fi(\))327 2257 y Fc(\000)p Fj(n)264 2300 y Fo(\000)g(!)55 b Fu(Coim\()p Fn(f)543 2283 y Fk(n)565 2300 y Fu(\)\).)37 b(Since)22 b(\()p Fn(f)820 2283 y Fg(0)815 2311 y Fk(n)838 2300 y Fn(\023)854 2307 y Fk(n)877 2300 y Fu(\))895 2283 y Fg(\000)p Fk(n)964 2300 y Fu(is)f(an)g(isomorphism,)i(w)o(e)e(ha)o(v) o(e)f(that)18 2370 y Fn(')48 2354 y Fg(0)72 2370 y Fu(:)12 b Fn(A)h Fo(\000)-7 b(!)12 b Fu(Coim\()p Fn(f)381 2354 y Fk(n)403 2370 y Fu(\))h(is)g(a)f(coimage)h(of)g Fn(f)761 2354 y Fk(n)796 2370 y Fu(as)g(w)o(ell)g(as)g(of)f Fn(')g Fu(=)h(\()p Fn(A)1217 2343 y Fk(')1239 2331 y Fc(0)1197 2370 y Fo(\000)-7 b(!)12 b Fu(Coim\()p Fn(f)1434 2354 y Fk(n)1456 2370 y Fu(\))h Fo(\000)-8 b(!)12 b Fn(A)p Fu(\).)18 2424 y(Th)o(us)17 b(Coim\()p Fn(')p Fu(\))f Fo(\000)-7 b(!)16 b Fn(A)h Fu(is)h(a)g(k)o(ernel)g(and)g(Ke\()p Fn(')p Fu(\))e(=)h(Ke\()p Fn(A)g Fo(\000)-8 b(!)16 b Fu(Coim\()p Fn(')p Fu(\)\))g(=)h(Ke\()p Fn(A)g Fo(\000)-8 b(!)611 2738 y Ff(krs.tex)23 b(-)g(Date:)47 b(June)24 b(23,)f(1994)g(Time:)47 b(10:45)p eop %%Page: 15 15 15 14 bop 644 54 a Fh(Krull-Sc)o(hmid)o(t)9 b(Theorem)578 b Fu(15)18 145 y(Coim\()p Fn(f)169 129 y Fk(n)191 145 y Fu(\)\))14 b(=)h(Ke)q(\()p Fn(f)392 129 y Fk(n)414 145 y Fu(\).)24 b(Since)18 b Fn(f)j Fu(is)c(b)q(ounded,)h Fn(')e Fu(satis\014es)h(\()p Fn(G)1148 152 y Fk(')1171 145 y Fu(\))f(and)h(Lemma)f(4.4)g(holds)18 199 y(for)e Fn(')p Fu(.)20 b(T)l(ranslated)15 b(bac)o(k)g(in)o(to)h(terms)e(of)h Fn(f)20 b Fu(giv)o(es)c(the)f(required)h(result.)p 1613 173 24 2 v 1613 198 2 25 v 1635 198 V 1613 200 24 2 v 18 353 a Fv(Example)i(4.7)g Fe(Consider)i(the)h(c)n(ate)n(gory)g(of)g (gr)n(oups)g Fa(Gr)o Fe(.)35 b(A)21 b(morphism)g Fn(f)26 b Fu(:)21 b Fn(G)g Fo(\000)-7 b(!)20 b Fn(G)18 407 y Fe(is)f(c)n(al)r(le)n(d)f(normal)i(if)f Fn(f)5 b Fu(\()p Fn(aba)518 390 y Fg(\000)p Fi(1)562 407 y Fu(\))18 b(=)h Fn(af)5 b Fu(\()p Fn(b)p Fu(\))p Fn(a)783 390 y Fg(\000)p Fi(1)846 407 y Fe(for)20 b(al)r(l)f Fn(a;)8 b(b)17 b Fo(2)i Fn(G)p Fe(.)30 b(If)19 b Fn(f)24 b Fe(is)19 b(normal)h(and)f Fn(G)18 461 y Fe(has)i(a.c.c.)36 b(and)21 b(d.c.c.)36 b(then)21 b(for)h(al)r(l)f Fn(n)h Fe(we)f(have)h Fn(f)995 444 y Fk(n)1017 461 y Fu(\()p Fn(aba)1103 444 y Fg(\000)p Fi(1)1147 461 y Fu(\))g(=)g Fn(af)1295 444 y Fk(n)1317 461 y Fu(\()p Fn(b)p Fu(\))p Fn(a)1397 444 y Fg(\000)p Fi(1)1462 461 y Fe(and)f(thus)18 515 y Fu(Coim\()p Fn(f)169 498 y Fk(n)191 515 y Fu(\))d(=)g(Im)q(\()p Fn(f)380 498 y Fk(n)402 515 y Fu(\))h Fe(normal)g(in)g Fn(G)p Fe(.)30 b(The)19 b(chains)f(of)i(sub)n(gr)n(oups)f Fu(Ke\()p Fn(f)1342 498 y Fk(n)1365 515 y Fu(\))f Fo(\022)g Fu(Ke)q(\()p Fn(f)1555 498 y Fk(n)p Fi(+1)1619 515 y Fu(\))18 569 y Fe(and)24 b Fu(Im\()p Fn(f)213 552 y Fk(n)236 569 y Fu(\))j Fo(\023)g Fu(Im)q(\()p Fn(f)443 552 y Fk(n)p Fi(+1)507 569 y Fu(\))d Fe(b)n(e)n(c)n(ome)f(stationary.)46 b(Final)r(ly)23 b Fn(a)k Fo(7!)g Fn(a)p Fu(\()p Fn(f)1326 552 y Fg(0)1321 580 y Fk(n)1343 569 y Fn(\023)1359 576 y Fk(n)1382 569 y Fu(\))1400 552 y Fg(\000)p Fi(1)1444 569 y Fn(f)1471 552 y Fk(n)1494 569 y Fu(\()p Fn(a)1536 552 y Fg(\000)p Fi(1)1580 569 y Fu(\))16 b Fo(\003)18 623 y Fu(\()p Fn(f)63 606 y Fg(0)58 634 y Fk(n)80 623 y Fn(\023)96 630 y Fk(n)119 623 y Fu(\))137 606 y Fg(\000)p Fi(1)181 623 y Fn(f)208 606 y Fk(n)231 623 y Fu(\()p Fn(a)p Fu(\))f Fe(is)h(a)g(se)n(ction)f(for)h(the)g(c)n(anonic)n(al)e (map)j Fu(Ke\()p Fn(f)1092 606 y Fk(n)1115 623 y Fu(\))1141 591 y Fm(`)1187 623 y Fu(Im\()p Fn(f)1286 606 y Fk(n)1308 623 y Fu(\))c Fo(\000)-8 b(!)13 b Fn(G)p Fe(.)20 b(Thus)c Fn(f)18 677 y Fe(is)g(b)n(ounde)n(d)g(and)g(the)g(Fitting)g(L)n(emma)g (holds.)461 806 y Fv(5)52 b(The)18 b(Krull-Sc)o(hmidt-Theorem)18 894 y(De\014nition)h(5.1)f Fu(Let)12 b Fn(A)h Fo(6)p Fu(=)g(0)e(b)q(e)i(in)f Fo(C)s Fu(.)19 b(W)l(e)12 b(call)h Fn(A)e Fe(inde)n(c)n(omp)n(osable)g Fu(if)h Fn(A)h Fu(=)g Fn(X)7 b Fo(\002)s Fn(Y)23 b Fu(implies)18 948 y Fn(X)16 b Fu(=)d(0)i(or)f Fn(Y)23 b Fu(=)13 b(0.)18 1054 y Fv(Lemma)k(5.2)h Fe(L)n(et)d Fn(A)h Fe(b)n(e)f(inde)n(c)n(omp)n(osable)g(and)h Fn(f)i Fu(:)12 b Fn(A)h Fo(\000)-8 b(!)12 b Fn(A)k Fe(a)g(b)n(ounde)n (d)g(endomorphism.)18 1108 y(Then)f Fn(f)22 b Fe(is)16 b(either)g(nilp)n(otent)f(or)i(an)f(automorphism.)76 1214 y Fd(Pro)q(of:)56 b Fu(By)19 b(the)g(Fitting)h(Lemma)f(there)g(is) h(an)f Fn(n)h Fo(2)g Fn(N)k Fu(suc)o(h)19 b(that)g Fn(A)g Fu(=)h(Ke)q(\()p Fn(f)1549 1197 y Fk(n)1571 1214 y Fu(\))12 b Fo(\002)18 1268 y Fu(Coim\()p Fn(f)169 1251 y Fk(n)191 1268 y Fu(\).)39 b(If)21 b(Coim\()p Fn(f)463 1251 y Fk(n)486 1268 y Fu(\))i(=)g(0)e(then)h Fn(f)766 1251 y Fk(n)812 1268 y Fu(=)h(0)e(and)h Fn(f)k Fu(is)c(nilp)q(oten)o(t.)40 b(If)22 b(Ke\()p Fn(f)1492 1251 y Fk(n)1515 1268 y Fu(\))h(=)g(0)18 1322 y(then)17 b(Coim\()p Fn(f)274 1305 y Fk(n)297 1322 y Fu(\))e Fo(\000)-7 b(!)15 b Fn(A)i Fu(m)o(ust)g(b)q(e)h(the)f(iden)o (tit)o(y)l(.)26 b(Hence)18 b Fn(f)1080 1305 y Fk(n)1119 1322 y Fu(:)d Fn(A)h Fo(\000)-7 b(!)15 b Fn(A)i Fu(has)g(k)o(ernel)h (zero)18 1376 y(and)g(is)h(a)f(di\013erence)h(cok)o(ernel.)30 b(By)19 b(axiom)f(\(I)q(I\))g(for)g Fo(C)j Fu(w)o(e)d(get)g(that)f Fn(f)1304 1359 y Fk(n)1345 1376 y Fu(and)h(also)h Fn(f)k Fu(are)18 1430 y(automorphisms.)p 1613 1403 V 1613 1428 2 25 v 1635 1428 V 1613 1430 24 2 v 18 1583 a Fv(De\014nition)c(5.3)f Fu(Giv)o(en)j Fn(A)i Fu(=)f Fn(B)16 b Fo(\002)e Fn(C)s Fu(.)37 b(W)l(e)21 b(de\014ne)h(the)f(subset)g Fn(X)26 b Fo(\022)d Fu(End\()p Fn(B)r Fu(\))d(of)h Fn(A)p Fu(-)18 1637 y Fe(pr)n(o)n(ductive)15 b Fu(endomorphisms)h(as)f(follo)o(ws:)55 1730 y(1.)j(If)d Fn(A)e Fu(=)g Fn(B)286 1714 y Fg(0)307 1730 y Fo(\002)e Fn(C)389 1714 y Fg(0)415 1730 y Fu(then)16 b(\()p Fn(B)e Fo(\000)-7 b(!)12 b Fn(A)h Fo(\000)-8 b(!)12 b Fn(B)841 1714 y Fg(0)865 1730 y Fo(\000)-7 b(!)12 b Fn(A)h Fo(\000)-7 b(!)12 b Fn(B)r Fu(\))g Fo(2)h Fn(X)t Fu(.)55 1824 y(2.)18 b(If)d Fn(f)r(;)8 b(g)14 b Fo(2)f Fn(X)t Fu(,)h(then)h Fn(f)5 b(g)15 b Fo(2)d Fn(X)t Fu(.)55 1917 y(3.)18 b(If)d Fn(f)j Fo(2)13 b Fn(X)g Fo(\\)e Fu(Aut\()p Fn(B)r Fu(\),)j(then)h Fn(f)636 1901 y Fg(\000)p Fi(1)693 1917 y Fo(2)e Fn(X)t Fu(.)55 2011 y(4.)18 b(If)d Fn(f)r(;)8 b(g)14 b Fo(2)f Fn(X)18 b Fu(are)d(summable)h(then)g Fn(f)f Fu(+)10 b Fn(g)k Fo(2)f Fn(X)t Fu(.)55 2104 y(5.)18 b Fn(X)e Fu(=)f(the)g(smallest)h(set)f(satisfying)h(\(1\))p Fn(;)8 b(:)g(:)f(:)e(;)j Fu(\(4\).)18 2210 y Fv(De\014nition)19 b(5.4)f Fn(A)12 b Fu(is)h Fe(b)n(ounde)n(d)f Fu(if)h(for)f(all)h Fn(A)g Fu(=)g Fn(B)7 b Fo(\002)t Fn(C)15 b Fu(all)e Fn(A)p Fu(-pro)q(ductiv)o(e)h(endomorphisms)18 2264 y Fn(f)20 b Fu(are)15 b(b)q(ounded.)18 2370 y Fv(Lemma)i(5.5)h Fe(L)n(et)i Fn(B)i Fe(b)n(e)d(inde)n(c)n(omp)n(osable)g(and)i Fn(A)f Fu(=)g Fn(B)15 b Fo(\002)e Fn(C)23 b Fe(b)n(e)d(b)n(ounde)n(d.) 32 b(L)n(et)20 b Fn(f)r(;)8 b(g)21 b Fe(b)n(e)18 2424 y Fn(A)p Fe(-pr)n(o)n(ductive)c(and)f(summable.)21 b(If)16 b Fn(f)21 b Fe(and)16 b Fn(g)i Fe(ar)n(e)e(nilp)n(otent,)f(then)h(so)g (is)g Fn(f)f Fu(+)c Fn(g)r Fe(.)611 2738 y Ff(krs.tex)23 b(-)g(Date:)47 b(June)24 b(23,)f(1994)g(Time:)47 b(10:45)p eop %%Page: 16 16 16 15 bop 18 54 a Fu(16)510 b Fh(Bo)q(do)11 b(P)o(areigis)f(and)g(Helm) o(ut)h(R\177)-18 b(ohrl)76 145 y Fd(Pro)q(of:)65 b Fu(Without)21 b(loss)h(of)f(generalit)o(y)h(w)o(e)f(assume)h Fn(f)28 b Fo(6)p Fu(=)c(0)e Fo(6)p Fu(=)i Fn(g)r Fu(.)38 b(Since)23 b Fn(f)d Fu(+)14 b Fn(g)23 b Fu(is)18 199 y Fn(A)p Fu(-pro)q(ductiv)o (e)18 b(it)f(is)h(b)q(ounded.)27 b(By)17 b(Lemma)g(5.2)f Fn(f)g Fu(+)c Fn(g)18 b Fu(is)g(either)g(nilp)q(oten)o(t)g(or)e(an)h (auto-)18 253 y(morphism.)23 b(Assume)16 b Fn(f)g Fu(+)11 b Fn(g)16 b Fo(2)e Fu(Aut\()p Fn(B)r Fu(\).)22 b(Then)17 b Fn(h)d Fu(=)h(\()p Fn(f)g Fu(+)c Fn(g)r Fu(\))1124 237 y Fg(\000)p Fi(1)1184 253 y Fu(is)16 b Fn(A)p Fu(-pro)q(ductiv)o(e) i(and)e(so)18 307 y(are)d Fn(hf)18 b Fu(and)c Fn(hg)r Fu(.)k(Th)o(us)c(id)f(=)g Fn(h)p Fu(\()p Fn(f)e Fu(+)6 b Fn(g)r Fu(\))12 b(=)h Fn(hf)f Fu(+)6 b Fn(hg)15 b Fu(with)f Fn(hf)k Fu(and)c Fn(hg)g(A)p Fu(-pro)q(ductiv)o(e,)h(hence)18 361 y(b)q(ounded.)20 b(Since)15 b Fn(f)i Fu(is)d(nilp)q(oten)o(t,)g (there)f(is)g(an)g Fn(n)g Fu(suc)o(h)g(that)f Fn(f)1106 345 y Fk(n)1141 361 y Fo(6)p Fu(=)h(0)g(and)g Fn(f)d Fo(\001)5 b Fn(f)1388 345 y Fk(n)1424 361 y Fu(=)13 b(0.)18 b(So)13 b Fn(f)1614 345 y Fk(n)18 415 y Fu(factors)f(through)h(Ke\()p Fn(f)5 b Fu(\))12 b Fo(\000)-7 b(!)12 b Fn(B)j Fu(and)f(w)o(e)e(get)h (Ke)q(\()p Fn(f)5 b Fu(\))12 b Fo(6)p Fu(=)h(0.)19 b(This)13 b(implies)j(that)c(Ke)q(\()p Fn(hf)5 b Fu(\))12 b Fo(6)p Fu(=)h(0)18 469 y(and)g(th)o(us)f Fn(hf)18 b Fu(is)13 b(not)f(an)h(automorphism.)19 b(By)12 b(Lemma)h(5.2)f Fn(hf)18 b Fu(is)13 b(then)g(nilp)q(oten)o(t.)20 b(Replac-)18 523 y(ing)e Fn(f)k Fu(b)o(y)17 b Fn(hf)23 b Fu(and)17 b Fn(g)i Fu(b)o(y)e Fn(hg)i Fu(w)o(e)e(can)g(assume)h(without)f(loss)g (of)g(generalit)o(y)h(that)f Fn(f)22 b Fu(and)17 b Fn(g)18 577 y Fu(are)e(summable,)g Fn(A)p Fu(-pro)q(ductiv)o(e)i(and)e(nilp)q (oten)o(t)h(with)g(id)d(=)g Fn(f)j Fu(+)10 b Fn(g)r Fu(.)76 631 y(Let)19 b Fn(f)188 614 y Fk(n)229 631 y Fu(=)f(0)g(=)g Fn(g)400 614 y Fk(n)422 631 y Fu(.)30 b(W)l(e)18 b(pro)o(v)o(e)g(b)o(y) h(induction)h(on)e Fn(k)i Fu(that)e Fn(f)1183 614 y Fk(t)1196 618 y Fb(1)1214 631 y Fn(g)r(f)1265 614 y Fk(t)1278 618 y Fb(2)1295 631 y Fn(g)9 b(:)f(:)g(:)d(g)r(f)1437 614 y Fk(t)1450 618 y Fj(k)1488 631 y Fu(=)18 b(0)g(for)18 685 y Fn(t)34 692 y Fk(i)64 685 y Fo(\025)e Fu(0)h(and)246 653 y Fm(P)290 663 y Fk(k)290 696 y(i)p Fi(=1)353 685 y Fn(t)369 692 y Fk(i)399 685 y Fu(=)g Fn(n)p Fu(.)26 b(F)l(or)16 b Fn(k)h Fu(=)g(1)g(this)g(is)h(trivial.)27 b(Using)18 b(Lemma)f(3.2)f(and)i(Lemma)18 739 y(3.4)12 b(the)i(induction)h(step)f(is)g Fn(f)532 722 y Fk(t)545 726 y Fb(1)563 739 y Fn(g)9 b(:)f(:)g(:)d(g)r(f)705 722 y Fk(t)718 726 y Fj(k)p Fb(+1)786 739 y Fu(=)13 b Fn(f)861 722 y Fk(t)874 726 y Fb(1)891 739 y Fn(g)c(:)f(:)g(:)e(g)r(f)1034 722 y Fk(t)1047 726 y Fj(k)1066 739 y Fn(f)f(f)1120 722 y Fk(t)1133 726 y Fj(k)p Fb(+1)1195 739 y Fu(+)i Fn(f)1264 722 y Fk(t)1277 726 y Fb(1)1295 739 y Fn(g)i(:)f(:)g(:)e(g)r(f)1438 722 y Fk(t)1451 726 y Fj(k)1470 739 y Fn(g)r(f)1521 722 y Fk(t)1534 726 y Fj(k)p Fb(+1)1601 739 y Fu(=)18 793 y Fn(f)45 776 y Fk(t)58 780 y Fb(1)76 793 y Fn(g)j(:)f(:)g(:)d(g)r(f) 218 776 y Fk(t)231 780 y Fj(k)251 793 y Fu(\()p Fn(f)17 b Fu(+)c Fn(g)r Fu(\))p Fn(f)425 776 y Fk(t)438 780 y Fj(k)p Fb(+1)511 793 y Fu(=)19 b Fn(f)592 776 y Fk(t)605 780 y Fb(1)623 793 y Fn(g)9 b(:)f(:)g(:)e(g)r(f)766 776 y Fk(t)779 780 y Fj(k)798 793 y Fn(f)825 776 y Fk(t)838 780 y Fj(k)p Fb(+1)912 793 y Fu(=)20 b(0)p Fn(:)e Fu(With)h(this)g (remark)g(w)o(e)f(get)h(\()p Fn(f)e Fu(+)18 847 y Fn(g)r Fu(\))60 830 y Fi(2)p Fk(n)113 847 y Fu(=)e(0)h(since)h(in)h(the)e (expansion)h(eac)o(h)g(summand)f(con)o(tains)h(at)f(least)g Fn(n)h Fu(factors)e Fn(f)22 b Fu(or)15 b Fn(n)18 901 y Fu(factors)f Fn(g)r Fu(,)g(so)h(it)g(is)h(zero.)k(Th)o(us)15 b Fn(f)g Fu(+)c Fn(g)j Fu(=)f(id)j(cannot)f(hold)h(with)f Fn(B)f Fo(6)p Fu(=)f(0.)p 1613 874 24 2 v 1613 899 2 25 v 1635 899 V 1613 901 24 2 v 18 1054 a Fv(Corollary)k(5.6)h Fe(L)n(et)13 b Fn(A)f Fu(=)h Fn(B)5 b Fo(\002)s Fn(C)16 b Fe(b)n(e)c(b)n(ounde)n(d)h(and)g Fn(B)i Fe(b)n(e)d(inde)n(c)n(omp)n (osable.)19 b(L)n(et)12 b Fn(f)1472 1061 y Fi(1)1491 1054 y Fn(;)c(:)g(:)g(:)d(;)j(f)1615 1061 y Fk(n)18 1108 y Fo(2)13 b Fu(End\()p Fn(B)r Fu(\))j Fe(b)n(e)g(summable)g(and)g Fn(A)p Fe(-pr)n(o)n(ductive.)22 b(If)16 b Fn(f)941 1115 y Fi(1)970 1108 y Fu(+)11 b Fn(:)d(:)g(:)g Fu(+)i Fn(f)1146 1115 y Fk(n)1182 1108 y Fu(=)j(id)k Fe(then)g(one)f(of)g(the)h Fn(f)1623 1115 y Fk(i)18 1162 y Fe(is)f(an)g(automorphism.)76 1244 y Fd(Pro)q(of:)51 b Fu(If)18 b(all)i Fn(f)387 1251 y Fk(i)419 1244 y Fu(are)e(nilp)q(oten)o(t)h(then)f(a)g(simple)i (induction)g(pro)q(of)e(sho)o(ws)f(that)g Fn(f)1570 1251 y Fi(1)1601 1244 y Fu(+)18 1298 y Fn(:)8 b(:)g(:)h Fu(+)i Fn(f)150 1305 y Fk(n)190 1298 y Fu(is)17 b(nilp)q(oten)o(t.)26 b(So)16 b(one)h(of)f(the)h Fn(f)758 1305 y Fk(i)789 1298 y Fu(cannot)f(b)q(e)i(nilp)q(oten)o(t,)f(and)g(hence)h(it)f(m)o(ust)f (b)q(e)18 1352 y(an)f(automorphism.)p 1613 1326 V 1613 1351 2 25 v 1635 1351 V 1613 1353 24 2 v 18 1506 a Fv(Theorem)i(5.7)h Fu(\(Krull-Sc)o(hmidt\))g Fe(L)n(et)471 1587 y Fn(A)13 b Fu(=)g Fn(A)600 1594 y Fi(1)629 1587 y Fo(\002)d Fn(:)e(:)g(:)g Fo(\002)j Fn(A)817 1594 y Fk(m)861 1587 y Fu(=)i Fn(B)944 1594 y Fi(1)972 1587 y Fo(\002)e Fn(:)d(:)g(:)g Fo(\002)i Fn(B)1161 1594 y Fk(n)18 1668 y Fe(b)n(e)16 b(two)h(de)n(c)n(omp)n (ositions)e(of)i Fn(A)g Fe(into)f(internal)g(dir)n(e)n(ct)g(pr)n(o)n (ducts)h(of)g(inde)n(c)n(omp)n(osable)e(sub)n(ob-)18 1722 y(je)n(cts)j Fn(A)158 1729 y Fk(i)190 1722 y Fe(r)n(esp.)28 b Fn(B)347 1729 y Fk(j)364 1722 y Fe(.)f(L)n(et)18 b Fn(A)h Fe(b)n(e)f(b)n(ounde)n(d.)28 b(Then)18 b Fn(m)e Fu(=)i Fn(n)h Fe(and)f Fn(A)1196 1729 y Fk(i)1227 1709 y Fo(\030)1227 1724 y Fu(=)1279 1722 y Fn(B)1314 1729 y Fk(i)1346 1722 y Fe(for)h(al)r(l)g Fn(i)f Fe(and)h(a)18 1776 y(suitable)d(r)n(e)n(or)n(dering)f(of)i(the)f Fn(B)562 1783 y Fk(i)575 1776 y Fe(s.)76 1858 y Fd(Pro)q(of:)39 b Fu(W)l(e)16 b(pro)o(v)o(e)e(the)h(follo)o(wing)h(statemen)o(t)f(b)o (y)g(induction)i(for)d Fn(t)f Fo(\024)g Fu(min)q(\()p Fn(m;)8 b(n)p Fu(\).)18 1912 y Fn(P)e Fu(\()p Fn(t)p Fu(\):)26 b(there)18 b(is)g(a)g(reordering)g(of)g Fn(B)662 1919 y Fi(1)680 1912 y Fn(;)8 b(:)g(:)g(:)d(;)j(B)817 1919 y Fk(n)856 1912 y Fu(suc)o(h)19 b(that)e Fn(A)1097 1919 y Fk(i)1128 1899 y Fo(\030)1128 1914 y Fu(=)1181 1912 y Fn(B)1216 1919 y Fk(i)1247 1912 y Fu(for)h Fn(i)f Fu(=)g(1)p Fn(;)8 b(:)g(:)g(:)d(;)j(t)18 b Fu(and)153 1966 y Fn(A)13 b Fu(=)g Fn(A)282 1973 y Fi(1)311 1966 y Fo(\002)d Fn(:)e(:)g(:)g Fo(\002)j Fn(A)499 1973 y Fk(t)523 1966 y Fo(\002)g Fn(B)604 1973 y Fk(t)p Fi(+1)670 1966 y Fo(\002)g Fn(:)d(:)g(:)g Fo(\002)i Fn(B)859 1973 y Fk(n)881 1966 y Fu(.)18 2020 y Fn(P)c Fu(\(0\))19 b(holds)h(b)o(y)f (h)o(yp)q(othesis.)33 b(Assume)20 b(that)f Fn(P)6 b Fu(\()p Fn(t)13 b Fo(\000)g Fu(1\))19 b(holds.)33 b(Then)20 b Fn(A)g Fu(=)g Fn(A)1456 2027 y Fi(1)1487 2020 y Fo(\002)13 b Fn(:)8 b(:)g(:)j Fo(\002)18 2073 y Fn(A)52 2080 y Fk(t)p Fg(\000)p Fi(1)120 2073 y Fo(\002)h Fn(B)202 2080 y Fk(t)228 2073 y Fo(\002)f Fn(:)d(:)g(:)i Fo(\002)h Fn(B)420 2080 y Fk(n)459 2073 y Fu(with)18 b(suitable)g(indexing)h(and)e Fn(A)1045 2080 y Fk(i)1075 2061 y Fo(\030)1075 2076 y Fu(=)1126 2073 y Fn(B)1161 2080 y Fk(i)1191 2073 y Fu(for)g(1)e Fo(\024)h Fn(i)g Fo(\024)g Fn(t)c Fo(\000)f Fu(1.)25 b(Let)18 2127 y Fn(p)41 2111 y Fg(0)41 2139 y Fk(i)70 2127 y Fu(and)17 b Fn(\023)176 2111 y Fg(0)176 2139 y Fk(i)206 2127 y Fu(b)q(e)f(the)g(corresp)q(onding)h(pro)s(jections)f (resp.)22 b(injections.)i(F)l(urthermore)15 b(w)o(e)h(ha)o(v)o(e)18 2181 y Fn(A)j Fu(=)g Fn(A)159 2188 y Fi(1)190 2181 y Fo(\002)13 b Fn(:)8 b(:)g(:)j Fo(\002)i Fn(A)386 2188 y Fk(m)437 2181 y Fu(with)19 b(the)g(pro)s(jections)g Fn(p)887 2188 y Fk(i)919 2181 y Fu(and)h(the)f(injections)h Fn(\023)1317 2188 y Fk(i)1331 2181 y Fu(.)31 b(Observ)o(e)19 b(that)18 2235 y Fn(\023)34 2242 y Fk(i)60 2235 y Fu(=)13 b Fn(\023)124 2219 y Fg(0)124 2247 y Fk(i)154 2235 y Fu(for)h Fn(i)f Fu(=)g(1)p Fn(;)8 b(:)g(:)g(:)t(;)g(t)i Fo(\000)g Fu(1,)15 b(and)g(that)g(w)o(e)g(use)g(the)h(same)f(sub)q(ob)s (jects)g Fn(A)1330 2242 y Fi(1)1349 2235 y Fn(;)8 b(:)g(:)g(:)t(;)g(A) 1484 2242 y Fk(t)p Fg(\000)p Fi(1)1541 2235 y Fu(,)15 b(but)18 2289 y(the)d(pro)s(jections)h(ma)o(y)e(b)q(e)j(di\013eren)o (t.)19 b(By)12 b(Prop)q(osition)h(3.5)f(w)o(e)g(ha)o(v)o(e)g(id)i(=)f Fn(\023)1338 2273 y Fg(0)1338 2301 y Fi(1)1356 2289 y Fn(p)1379 2273 y Fg(0)1379 2301 y Fi(1)1402 2289 y Fu(+)5 b Fn(:)j(:)g(:)s Fu(+)d Fn(\023)1556 2273 y Fg(0)1556 2301 y Fk(n)1579 2289 y Fn(p)1602 2273 y Fg(0)1602 2301 y Fk(n)1624 2289 y Fu(.)18 2343 y(Since)17 b Fn(p)160 2350 y Fk(t)174 2343 y Fn(\023)190 2327 y Fg(0)190 2355 y Fk(i)217 2343 y Fu(=)c Fn(p)288 2350 y Fk(t)302 2343 y Fn(\023)318 2350 y Fk(i)345 2343 y Fu(=)g(0)h(for)h(1)d Fo(\024)h Fn(i)g Fo(\024)g Fn(t)d Fo(\000)h Fu(1)j(b)o(y)i (de\014nition)h(of)d(the)i(injections,)g(w)o(e)f(get)109 2424 y(id)147 2431 y Fk(A)172 2435 y Fj(t)200 2424 y Fu(=)e Fn(p)271 2431 y Fk(t)286 2424 y Fn(\023)302 2431 y Fk(t)329 2424 y Fu(=)g Fn(p)400 2431 y Fk(t)422 2424 y Fu(id)8 b Fn(\023)483 2431 y Fk(t)511 2424 y Fu(=)13 b Fn(p)582 2431 y Fk(t)596 2424 y Fn(\023)612 2405 y Fg(0)612 2435 y Fi(1)631 2424 y Fn(p)654 2405 y Fg(0)654 2435 y Fi(1)672 2424 y Fn(\023)688 2431 y Fk(t)713 2424 y Fu(+)e Fn(:)d(:)g(:)g Fu(+)i Fn(p)890 2431 y Fk(t)905 2424 y Fn(\023)921 2405 y Fg(0)921 2435 y Fk(n)943 2424 y Fn(p)966 2405 y Fg(0)966 2435 y Fk(n)989 2424 y Fn(\023)1005 2431 y Fk(t)1032 2424 y Fu(=)j Fn(p)1103 2431 y Fk(t)1118 2424 y Fn(\023)1134 2405 y Fg(0)1134 2435 y Fk(t)1148 2424 y Fn(p)1171 2405 y Fg(0)1171 2435 y Fk(t)1186 2424 y Fn(\023)1202 2431 y Fk(t)1227 2424 y Fu(+)d Fn(:)e(:)g(:)g Fu(+)j Fn(p)1404 2431 y Fk(t)1418 2424 y Fn(\023)1434 2405 y Fg(0)1434 2435 y Fk(n)1457 2424 y Fn(p)1480 2405 y Fg(0)1480 2435 y Fk(n)1502 2424 y Fn(\023)1518 2431 y Fk(t)1533 2424 y Fn(:)611 2738 y Ff(krs.tex)23 b(-)g(Date:)47 b(June)24 b(23,)f(1994)g(Time:)47 b(10:45)p eop %%Page: 17 17 17 16 bop 644 54 a Fh(Krull-Sc)o(hmid)o(t)9 b(Theorem)578 b Fu(17)18 145 y(By)11 b(the)g(Corollary)g(5.6)g(one)g(of)g(the)g Fn(p)650 152 y Fk(t)665 145 y Fn(\023)681 129 y Fg(0)681 157 y Fk(i)695 145 y Fn(p)718 129 y Fg(0)718 157 y Fk(i)731 145 y Fn(\023)747 152 y Fk(t)773 145 y Fu(m)o(ust)g(b)q(e)h(an)f (automorphism.)18 b(After)11 b(reindexing)18 199 y(w)o(e)22 b(ma)o(y)g(assume)g(that)f Fn(p)490 206 y Fk(t)505 199 y Fn(\023)521 183 y Fg(0)521 210 y Fk(t)536 199 y Fn(p)559 183 y Fg(0)559 210 y Fk(t)573 199 y Fn(\023)589 206 y Fk(t)626 199 y Fu(is)i(an)f(automorphism)g(of)g Fn(A)1150 206 y Fk(t)1165 199 y Fu(.)41 b(Since)24 b(\()p Fn(p)1386 206 y Fk(t)1400 199 y Fn(\023)1416 183 y Fg(0)1416 210 y Fk(t)1431 199 y Fn(p)1454 183 y Fg(0)1454 210 y Fk(t)1468 199 y Fn(\023)1484 206 y Fk(t)1499 199 y Fu(\))1517 183 y Fk(r)q Fi(+1)1601 199 y Fu(=)18 253 y Fn(p)41 260 y Fk(t)55 253 y Fn(\023)71 237 y Fg(0)71 264 y Fk(t)86 253 y Fu(\()p Fn(p)127 237 y Fg(0)127 264 y Fk(t)141 253 y Fn(\023)157 260 y Fk(t)172 253 y Fn(p)195 260 y Fk(t)209 253 y Fn(\023)225 237 y Fg(0)225 264 y Fk(t)240 253 y Fu(\))258 237 y Fk(r)276 253 y Fn(p)299 237 y Fg(0)299 264 y Fk(t)314 253 y Fn(\023)330 260 y Fk(t)360 253 y Fu(is)17 b(also)f(an)g(automorphism,)f(w)o(e)h(get)g(\()p Fn(p)1063 237 y Fg(0)1063 264 y Fk(t)1077 253 y Fn(\023)1093 260 y Fk(t)1108 253 y Fn(p)1131 260 y Fk(t)1145 253 y Fn(\023)1161 237 y Fg(0)1161 264 y Fk(t)1176 253 y Fu(\))1194 237 y Fk(r)1226 253 y Fo(6)p Fu(=)e(0,)i(hence)h Fn(p)1477 237 y Fg(0)1477 264 y Fk(t)1491 253 y Fn(\023)1507 260 y Fk(t)1522 253 y Fn(p)1545 260 y Fk(t)1560 253 y Fn(\023)1576 237 y Fg(0)1576 264 y Fk(t)1606 253 y Fu(is)18 307 y(also)12 b(an)f(automorphism)h(due)g(to)f(Lemma)h(5.2.)18 b(Th)o(us)11 b Fn(p)980 291 y Fg(0)980 318 y Fk(t)995 307 y Fn(\023)1011 314 y Fk(t)1038 307 y Fu(:)i Fn(A)1098 314 y Fk(t)1125 307 y Fo(\000)-7 b(!)12 b Fn(B)1246 314 y Fk(t)1272 307 y Fu(is)g(an)g(isomorphism.)76 361 y(It)17 b(remains)h(to)f(sho)o(w)g (that)g Fn(A)f Fu(=)h Fn(A)709 368 y Fi(1)739 361 y Fo(\002)12 b Fn(:)c(:)g(:)i Fo(\002)i Fn(A)932 368 y Fk(t)958 361 y Fo(\002)g Fn(B)1040 368 y Fk(t)p Fi(+1)1108 361 y Fo(\002)g Fn(:)c(:)g(:)i Fo(\002)i Fn(B)1302 368 y Fk(n)1324 361 y Fu(.)26 b(W)l(e)18 b(ha)o(v)o(e)f Fn(A)f Fu(=)18 415 y Fn(B)53 422 y Fk(t)73 415 y Fo(\002)6 b Fu(\()p Fn(A)166 422 y Fi(1)191 415 y Fo(\002)g Fn(:)i(:)g(:)d Fo(\002)h Fn(A)367 422 y Fk(t)p Fg(\000)p Fi(1)431 415 y Fo(\002)g Fn(B)507 422 y Fk(t)p Fi(+1)570 415 y Fo(\002)g Fn(:)i(:)g(:)t Fo(\002)e Fn(B)746 422 y Fk(n)769 415 y Fu(\).)19 b(Let)13 b Fn(X)j Fu(=)d Fn(A)1034 422 y Fi(1)1059 415 y Fo(\002)6 b Fn(:)i(:)g(:)d Fo(\002)h Fn(A)1235 422 y Fk(t)p Fg(\000)p Fi(1)1299 415 y Fo(\002)g Fn(B)1375 422 y Fk(t)p Fi(+1)1437 415 y Fo(\002)g Fn(:)i(:)g(:)d Fo(\002)h Fn(B)1614 422 y Fk(n)18 469 y Fu(as)18 b(sub)q(ob)s(ject)h(of)f Fn(A)p Fu(.)31 b(Then)19 b Fn(A)577 476 y Fk(t)610 469 y Fu(is)g(a)g (complemen)o(t)g(for)f Fn(X)k Fu(in)d Fn(A)g Fu(=)g Fn(B)1289 476 y Fk(t)1315 469 y Fo(\002)13 b Fn(X)22 b Fu(since)e Fn(p)1561 453 y Fg(0)1561 480 y Fk(t)1575 469 y Fn(\023)1591 476 y Fk(t)1624 469 y Fu(:)18 523 y Fn(A)52 530 y Fk(t)82 523 y Fo(\000)-7 b(!)15 b Fn(A)g Fo(\000)-7 b(!)15 b Fn(B)344 530 y Fk(t)375 523 y Fu(is)i(an)g(isomorphism.)25 b(By)17 b(Prop)q(osition)g(2.6)f(w)o(e)h(get)f Fn(A)1349 530 y Fk(t)1375 523 y Fo(\\)c Fn(X)18 b Fu(=)e(0)h(and)18 577 y Fn(A)52 584 y Fk(t)79 577 y Fo([)c Fn(X)22 b Fu(=)c Fn(A)p Fu(.)31 b(F)l(urthermore)18 b Fn(A)614 584 y Fk(t)648 577 y Fo(\000)-8 b(!)18 b Fn(A)h Fu(and)g Fn(X)j Fo(\000)-7 b(!)18 b Fn(A)g Fu(are)h(k)o(ernels.)31 b(Hence)20 b Fn(A)f Fu(is)g(an)18 631 y(in)o(ternal)d(direct)g(pro)q(duct)f Fn(A)e Fu(=)g Fn(A)614 638 y Fk(t)639 631 y Fo(\002)d Fn(X)19 b Fu(of)14 b Fn(A)826 638 y Fk(t)856 631 y Fu(and)i Fn(X)t Fu(.)p 1613 604 24 2 v 1613 629 2 25 v 1635 629 V 1613 631 24 2 v 705 785 a Fv(References)37 859 y Fr(1.)38 b(P)o(aul)14 b(M.)f(Cohn:)k Ft(Universal)11 b(A)o(lgebr)n(a.)g Fr(2nd)j(ed.,)e(Reidel,)j(Dordrec)o(h)o(t,)e(1981.)37 901 y(2.)38 b(Bo)q(do)14 b(P)o(areigis:)19 b Ft(Cate)n(gories)12 b(and)g(F)m(unctors.)f Fr(Academic)j(Press,)f(1970.)37 942 y(3.)38 b(G)q(\177)-20 b(un)o(ther)13 b(Ric)o(h)o(ter:)18 b Ft(Krul)r(l-Schmidt)11 b(for)i(arbitr)n(ary)e(c)n(ate)n(gories.)e Fr(Con)o(tributions)15 b(to)d(General)i(Algebra,)105 984 y(2)f(\(Klagenfurt,)h(1982\),)f(319-342,)h(H\177)-19 b(older-Pic)o(hler-T)m(empsky)m(,)15 b(Vienna,)f(1983.)611 2738 y Ff(krs.tex)23 b(-)g(Date:)47 b(June)24 b(23,)f(1994)g(Time:)47 b(10:45)p eop %%Trailer end userdict /end-hook 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