251:
517:
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978:
925:
803:
720:
349:
421:
666:
639:
279:
391:
588:
448:
305:
744:
608:
561:
541:
308:
201:
1019:
453:
1103:
815:
930:
877:
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1133:
1128:
314:
984:
673:
58:
282:
123:
139:
1123:
396:
723:
641:
is an abelian category by using localization theory (also Swan). This is the hard part of the proof.
108:
50:
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620:
260:
1041:
1074:
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78:
354:
17:
1099:
1015:
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175:
46:
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1007:
179:
155:
131:
89:
45:; it essentially states that these categories, while rather abstractly defined, are in fact
42:
566:
426:
54:
287:
27:
Abelian categories, while abstractly defined, are in fact concrete categories of modules
988:
729:
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171:
1117:
1031:
93:
669:
520:
85:
62:
1095:
246:{\displaystyle {\mathcal {L}}\subset \operatorname {Fun} ({\mathcal {A}},Ab)}
174:
and sums of morphisms being determined as in the case of modules. However,
143:
1078:
1011:
1070:
874:
we get another contravariant, exact and fully faithful embedding
512:{\displaystyle h^{A}(X)=\operatorname {Hom} _{\mathcal {A}}(A,X)}
1033:
Abelian
Categories: An Introduction to the Theory of Functors
980:
is the desired covariant exact and fully faithful embedding.
867:{\displaystyle G(B)=\operatorname {Hom} _{\mathcal {L}}(B,I)}
150:
correspond to the ordinary kernels and cokernels computed in
945:
889:
840:
771:
695:
653:
626:
485:
408:
336:
326:
266:
226:
207:
1057:
Mitchell, Barry (July 1964). "The Full
Imbedding Theorem".
973:{\displaystyle GH:{\mathcal {A}}\to R\operatorname {-Mod} }
920:{\displaystyle G:{\mathcal {L}}\to R\operatorname {-Mod} .}
798:{\displaystyle R:=\operatorname {Hom} _{\mathcal {L}}(I,I)}
590:
is already left exact. The proof of the right exactness of
186:
do not necessarily correspond to projective and injective
543:
is fully faithful and we also get the left exactness of
158:. The theorem thus essentially says that the objects of
57:
proofs in these categories. The theorem is named after
933:
880:
818:
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732:
715:{\displaystyle \bigoplus _{A\in {\mathcal {A}}}h^{A}}
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1004:
107:-Mod (where the latter denotes the category of all
1065:(3). The Johns Hopkins University Press: 619–637.
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385:
344:{\displaystyle H:{\mathcal {A}}\to {\mathcal {L}}}
343:
299:
273:
245:
1046:Reprints in Theory and Applications of Categories
84:(with 1, not necessarily commutative) and a
77:is a small abelian category, then there exists a
644:It is easy to check that the abelian category
1090:. Cambridge Studies in Advanced Mathematics.
8:
726:and therefore has an injective cogenerator
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154:-Mod. Such an equivalence is necessarily
805:is the ring we need for the category of
73:The precise statement is as follows: if
170:-linear maps, with kernels, cokernels,
1088:An introduction to homological algebra
53:. This allows one to use element-wise
7:
610:is harder and can be read in Swan,
416:{\displaystyle A\in {\mathcal {A}}}
966:
963:
960:
910:
907:
904:
25:
985:Gabriel–Quillen embedding theorem
18:Freyd–Mitchell embedding theorem
1059:American Journal of Mathematics
612:Lecture Notes in Mathematics 76
166:-modules, and the morphisms as
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894:
861:
849:
828:
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661:{\displaystyle {\mathcal {L}}}
634:{\displaystyle {\mathcal {L}}}
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450:is the covariant hom-functor,
367:
361:
331:
274:{\displaystyle {\mathcal {A}}}
240:
221:
1:
1039:reprinted with a forward as
31:Mitchell's embedding theorem
983:Note that the proof of the
1150:
1086:Charles A. Weibel (1993).
386:{\displaystyle H(A)=h^{A}}
283:category of abelian groups
257:from the abelian category
722:. In other words it is a
617:After that we prove that
1096:10.1017/CBO9781139644136
138:-Mod in such a way that
307:. First we construct a
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39:full embedding theorem
35:Freyd–Mitchell theorem
991:is almost identical.
975:
922:
869:
800:
741:
724:Grothendieck category
717:
663:
636:
605:
585:
583:{\displaystyle h^{A}}
558:
538:
514:
445:
443:{\displaystyle h^{A}}
418:
388:
346:
302:
276:
248:
162:can be thought of as
1042:"Abelian Categories"
1029:Peter Freyd (1964).
931:
878:
816:
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730:
679:
648:
621:
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563:very easily because
547:
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454:
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288:
261:
202:
41:, is a result about
33:, also known as the
1134:Theorems in algebra
1129:Additive categories
1002:R. G. Swan (1968).
255:left exact functors
253:be the category of
194:Sketch of the proof
47:concrete categories
1012:10.1007/BFb0080281
970:
917:
864:
795:
736:
712:
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658:
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553:
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341:
300:{\displaystyle Ab}
297:
271:
243:
43:abelian categories
1037:. Harper and Row.
1021:978-3-540-04245-7
959:
903:
751:endomorphism ring
739:{\displaystyle I}
682:
603:{\displaystyle H}
556:{\displaystyle H}
536:{\displaystyle H}
16:(Redirected from
1141:
1109:
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1038:
1036:
1025:
989:exact categories
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927:The composition
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132:full subcategory
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1071:10.2307/2373027
1056:
1052:: 23–164. 2003.
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172:exact sequences
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55:diagram chasing
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59:Barry Mitchell
26:
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6:
4:
3:
2:
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1124:Module theory
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1105:9781139644136
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309:contravariant
294:
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237:
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98:
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94:exact functor
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68:
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56:
52:
48:
44:
40:
36:
32:
19:
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1062:
1058:
1049:
1045:
1032:
1006:. Springer.
1003:
982:
811:
806:
748:
670:AB5 category
643:
616:
611:
523:states that
521:Yoneda Lemma
197:
187:
183:
167:
163:
159:
151:
147:
146:computed in
135:
127:
119:
118:The functor
117:
110:
104:
100:
96:
81:
74:
72:
38:
34:
30:
29:
182:objects in
124:equivalence
63:Peter Freyd
1118:Categories
995:References
809:-modules.
311:embedding
190:-modules.
176:projective
122:yields an
951:→
895:→
847:
778:
691:∈
684:⨁
674:generator
492:
404:∈
332:→
219:
213:⊂
180:injective
144:cokernels
423:, where
393:for all
156:additive
126:between
113:-modules
90:faithful
1079:2373027
672:with a
281:to the
140:kernels
69:Details
51:modules
37:or the
1102:
1077:
1018:
668:is an
519:. The
130:and a
1075:JSTOR
109:left
1100:ISBN
1016:ISBN
987:for
749:The
198:Let
178:and
142:and
92:and
86:full
79:ring
61:and
1092:doi
1067:doi
1008:doi
836:Hom
812:By
767:Hom
481:Hom
351:by
216:Fun
134:of
115:).
49:of
1120::
1098:.
1073:.
1063:86
1061:.
1048:.
1044:.
1014:.
763::=
746:.
614:.
103:→
99::
88:,
65:.
1108:.
1094::
1081:.
1069::
1050:3
1024:.
1010::
967:d
964:o
961:M
958:-
954:R
946:A
941::
938:H
935:G
915:.
911:d
908:o
905:M
902:-
898:R
890:L
885::
882:G
862:)
859:I
856:,
853:B
850:(
841:L
832:=
829:)
826:B
823:(
820:G
807:R
793:)
790:I
787:,
784:I
781:(
772:L
760:R
734:I
708:A
704:h
696:A
688:A
654:L
627:L
598:H
576:A
572:h
551:H
531:H
507:)
504:X
501:,
498:A
495:(
486:A
477:=
474:)
471:X
468:(
463:A
459:h
436:A
432:h
409:A
401:A
379:A
375:h
371:=
368:)
365:A
362:(
359:H
337:L
327:A
322::
319:H
295:b
292:A
267:A
241:)
238:b
235:A
232:,
227:A
222:(
208:L
188:R
184:A
168:R
164:R
160:A
152:R
148:A
136:R
128:A
120:F
111:R
105:R
101:A
97:F
82:R
75:A
20:)
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