240:
506:
861:
967:
914:
792:
709:
338:
410:
655:
628:
268:
380:
577:
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294:
733:
597:
550:
530:
297:
190:
1008:
442:
1092:
804:
919:
866:
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1122:
1117:
303:
973:
662:
47:
271:
112:
128:
1112:
385:
712:
630:
is an abelian category by using localization theory (also Swan). This is the hard part of the proof.
97:
39:
636:
609:
249:
1030:
1063:
243:
67:
343:
1088:
1004:
739:
164:
35:
1080:
1055:
996:
168:
144:
120:
78:
34:; it essentially states that these categories, while rather abstractly defined, are in fact
31:
555:
415:
43:
276:
16:
Abelian categories, while abstractly defined, are in fact concrete categories of modules
977:
718:
582:
535:
515:
160:
1106:
1020:
82:
658:
509:
74:
51:
1084:
235:{\displaystyle {\mathcal {L}}\subset \operatorname {Fun} ({\mathcal {A}},Ab)}
163:
and sums of morphisms being determined as in the case of modules. However,
132:
1067:
1000:
1059:
863:
we get another contravariant, exact and fully faithful embedding
501:{\displaystyle h^{A}(X)=\operatorname {Hom} _{\mathcal {A}}(A,X)}
1022:
Abelian
Categories: An Introduction to the Theory of Functors
969:
is the desired covariant exact and fully faithful embedding.
856:{\displaystyle G(B)=\operatorname {Hom} _{\mathcal {L}}(B,I)}
139:
correspond to the ordinary kernels and cokernels computed in
934:
878:
829:
760:
684:
642:
615:
474:
397:
325:
315:
255:
215:
196:
1046:
Mitchell, Barry (July 1964). "The Full
Imbedding Theorem".
962:{\displaystyle GH:{\mathcal {A}}\to R\operatorname {-Mod} }
909:{\displaystyle G:{\mathcal {L}}\to R\operatorname {-Mod} .}
787:{\displaystyle R:=\operatorname {Hom} _{\mathcal {L}}(I,I)}
579:
is already left exact. The proof of the right exactness of
175:
do not necessarily correspond to projective and injective
532:
is fully faithful and we also get the left exactness of
147:. The theorem thus essentially says that the objects of
46:
proofs in these categories. The theorem is named after
922:
869:
807:
747:
721:
704:{\displaystyle \bigoplus _{A\in {\mathcal {A}}}h^{A}}
670:
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445:
418:
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346:
306:
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193:
993:
96:-Mod (where the latter denotes the category of all
1054:(3). The Johns Hopkins University Press: 619–637.
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374:
333:{\displaystyle H:{\mathcal {A}}\to {\mathcal {L}}}
332:
288:
262:
234:
1035:Reprints in Theory and Applications of Categories
73:(with 1, not necessarily commutative) and a
66:is a small abelian category, then there exists a
633:It is easy to check that the abelian category
1079:. Cambridge Studies in Advanced Mathematics.
8:
715:and therefore has an injective cogenerator
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143:-Mod. Such an equivalence is necessarily
794:is the ring we need for the category of
62:The precise statement is as follows: if
159:-linear maps, with kernels, cokernels,
1077:An introduction to homological algebra
42:. This allows one to use element-wise
7:
599:is harder and can be read in Swan,
405:{\displaystyle A\in {\mathcal {A}}}
955:
952:
949:
899:
896:
893:
14:
974:Gabriel–Quillen embedding theorem
1048:American Journal of Mathematics
601:Lecture Notes in Mathematics 76
155:-modules, and the morphisms as
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883:
850:
838:
817:
811:
781:
769:
650:{\displaystyle {\mathcal {L}}}
623:{\displaystyle {\mathcal {L}}}
495:
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456:
439:is the covariant hom-functor,
356:
350:
320:
263:{\displaystyle {\mathcal {A}}}
229:
210:
1:
1028:reprinted with a forward as
20:Mitchell's embedding theorem
972:Note that the proof of the
1139:
1075:Charles A. Weibel (1993).
375:{\displaystyle H(A)=h^{A}}
272:category of abelian groups
246:from the abelian category
711:. In other words it is a
606:After that we prove that
1085:10.1017/CBO9781139644136
127:-Mod in such a way that
296:. First we construct a
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406:
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290:
264:
236:
28:full embedding theorem
24:Freyd–Mitchell theorem
980:is almost identical.
964:
911:
858:
789:
730:
713:Grothendieck category
706:
652:
625:
594:
574:
572:{\displaystyle h^{A}}
547:
527:
503:
434:
432:{\displaystyle h^{A}}
407:
377:
335:
291:
265:
237:
151:can be thought of as
1031:"Abelian Categories"
1018:Peter Freyd (1964).
920:
867:
805:
745:
719:
668:
637:
610:
583:
556:
552:very easily because
536:
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443:
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344:
304:
277:
250:
191:
30:, is a result about
22:, also known as the
1123:Theorems in algebra
1118:Additive categories
991:R. G. Swan (1968).
244:left exact functors
242:be the category of
183:Sketch of the proof
36:concrete categories
1001:10.1007/BFb0080281
959:
906:
853:
784:
725:
701:
690:
647:
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589:
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522:
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429:
402:
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330:
289:{\displaystyle Ab}
286:
260:
232:
32:abelian categories
1026:. Harper and Row.
1010:978-3-540-04245-7
948:
892:
740:endomorphism ring
728:{\displaystyle I}
671:
592:{\displaystyle H}
545:{\displaystyle H}
525:{\displaystyle H}
1130:
1098:
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978:exact categories
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916:The composition
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121:full subcategory
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1103:
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1060:10.2307/2373027
1045:
1041:: 23–164. 2003.
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161:exact sequences
60:
44:diagram chasing
17:
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59:
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48:Barry Mitchell
15:
13:
10:
9:
6:
4:
3:
2:
1135:
1124:
1121:
1119:
1116:
1114:
1113:Module theory
1111:
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1108:
1096:
1094:9781139644136
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451:
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392:
389:
367:
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347:
310:
307:
299:
298:contravariant
283:
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273:
245:
226:
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95:
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83:exact functor
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55:
53:
49:
45:
41:
37:
33:
29:
25:
21:
1076:
1051:
1047:
1038:
1034:
1021:
995:. Springer.
992:
971:
800:
795:
737:
659:AB5 category
632:
605:
600:
512:states that
510:Yoneda Lemma
186:
176:
172:
156:
152:
148:
140:
136:
135:computed in
124:
116:
108:
107:The functor
106:
99:
93:
89:
85:
70:
63:
61:
27:
23:
19:
18:
171:objects in
113:equivalence
52:Peter Freyd
1107:Categories
984:References
798:-modules.
300:embedding
179:-modules.
165:projective
111:yields an
940:→
884:→
836:
767:
680:∈
673:⨁
663:generator
481:
393:∈
321:→
208:
202:⊂
169:injective
133:cokernels
412:, where
382:for all
145:additive
115:between
102:-modules
79:faithful
1068:2373027
661:with a
270:to the
129:kernels
58:Details
40:modules
26:or the
1091:
1066:
1007:
657:is an
508:. The
119:and a
1064:JSTOR
98:left
1089:ISBN
1005:ISBN
976:for
738:The
187:Let
167:and
131:and
81:and
75:full
68:ring
50:and
1081:doi
1056:doi
997:doi
825:Hom
801:By
756:Hom
470:Hom
340:by
205:Fun
123:of
104:).
38:of
1109::
1087:.
1062:.
1052:86
1050:.
1037:.
1033:.
1003:.
752::=
735:.
603:.
92:→
88::
77:,
54:.
1097:.
1083::
1070:.
1058::
1039:3
1013:.
999::
956:d
953:o
950:M
947:-
943:R
935:A
930::
927:H
924:G
904:.
900:d
897:o
894:M
891:-
887:R
879:L
874::
871:G
851:)
848:I
845:,
842:B
839:(
830:L
821:=
818:)
815:B
812:(
809:G
796:R
782:)
779:I
776:,
773:I
770:(
761:L
749:R
723:I
697:A
693:h
685:A
677:A
643:L
616:L
587:H
565:A
561:h
540:H
520:H
496:)
493:X
490:,
487:A
484:(
475:A
466:=
463:)
460:X
457:(
452:A
448:h
425:A
421:h
398:A
390:A
368:A
364:h
360:=
357:)
354:A
351:(
348:H
326:L
316:A
311::
308:H
284:b
281:A
256:A
230:)
227:b
224:A
221:,
216:A
211:(
197:L
177:R
173:A
157:R
153:R
149:A
141:R
137:A
125:R
117:A
109:F
100:R
94:R
90:A
86:F
71:R
64:A
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