162:
764:
709:
573:
413:
1069:
497:
1111:
1033:
1009:
985:
954:
930:
894:
653:
629:
597:
521:
445:
379:
355:
328:
304:
230:
186:
97:
66:
280:
255:
831:
1131:
855:
808:
784:
206:
717:
662:
526:
956:
is closed under arbitrary coproducts (a.k.a. direct sums). Hence the notion of a localizing subcategory is equivalent to the notion of a hereditary
113:
448:
1183:
384:
1038:
866:
466:
1178:
1084:
1075:
787:
456:
905:
32:. Localizing subcategories are certain Serre subcategories. They are strongly linked to the notion of a
1014:
990:
966:
935:
911:
875:
634:
610:
578:
502:
426:
360:
336:
309:
285:
211:
167:
78:
47:
1079:
1146:
1134:
897:
460:
423:
between abelian categories are Serre subcategories, and that one can build (for locally small
33:
712:
69:
29:
1158:
260:
813:
235:
1116:
957:
901:
840:
793:
769:
452:
191:
108:
1172:
862:
420:
416:
73:
25:
759:{\displaystyle S\colon {\mathcal {A}}/{\mathcal {C}}\rightarrow {\mathcal {A}}}
704:{\displaystyle T\colon {\mathcal {A}}\rightarrow {\mathcal {A}}/{\mathcal {C}}}
568:{\displaystyle T\colon {\mathcal {A}}\rightarrow {\mathcal {A}}/{\mathcal {C}}}
523:, is abelian, and comes with an exact functor (called the quotient functor)
157:{\displaystyle 0\rightarrow A'\rightarrow A\rightarrow A''\rightarrow 0}
1078:
implies that every
Grothendieck category is the quotient category of a
790:, each localizing subcategory is closed under colimits. The functor
419:. The importance of this notion stems from the fact that kernels of
330:
is closed under subobjects, quotient objects and extensions.
1056:
1044:
1020:
996:
972:
941:
917:
881:
751:
741:
729:
696:
684:
674:
640:
616:
584:
560:
548:
538:
508:
484:
472:
432:
400:
390:
366:
342:
315:
291:
217:
173:
84:
53:
1163:
Abelian
Categories with Applications to Rings and Modules
381:
is itself an abelian category, and the inclusion functor
1119:
1087:
1041:
1017:
993:
969:
938:
914:
878:
843:
816:
796:
772:
720:
665:
637:
613:
581:
529:
505:
469:
429:
387:
363:
339:
312:
288:
263:
238:
214:
194:
170:
116:
81:
50:
1125:
1105:
1063:
1027:
1003:
979:
948:
924:
888:
849:
825:
802:
778:
758:
703:
647:
623:
591:
567:
515:
491:
439:
407:
373:
349:
322:
298:
274:
249:
224:
200:
180:
156:
91:
60:
408:{\displaystyle {\mathcal {C}}\to {\mathcal {A}}}
1064:{\displaystyle {\mathcal {A}}/{\mathcal {C}}}
492:{\displaystyle {\mathcal {A}}/{\mathcal {C}}}
8:
1118:
1086:
1055:
1054:
1049:
1043:
1042:
1040:
1019:
1018:
1016:
995:
994:
992:
971:
970:
968:
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939:
937:
916:
915:
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842:
815:
795:
771:
750:
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740:
739:
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719:
695:
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664:
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583:
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477:
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431:
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193:
172:
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115:
83:
82:
80:
52:
51:
49:
631:be locally small. The Serre subcategory
1106:{\displaystyle \operatorname {Mod} (R)}
1165:; Academic Press, Inc.; out of print.
7:
1137:) modulo a localizing subcategory.
1071:are again Grothendieck categories.
14:
499:, which has the same objects as
1011:a localizing subcategory, then
987:is a Grothendieck category and
786:, as a left adjoint, preserves
1100:
1094:
1028:{\displaystyle {\mathcal {C}}}
1004:{\displaystyle {\mathcal {C}}}
980:{\displaystyle {\mathcal {A}}}
949:{\displaystyle {\mathcal {C}}}
925:{\displaystyle {\mathcal {C}}}
889:{\displaystyle {\mathcal {A}}}
746:
679:
648:{\displaystyle {\mathcal {C}}}
624:{\displaystyle {\mathcal {A}}}
592:{\displaystyle {\mathcal {C}}}
543:
516:{\displaystyle {\mathcal {A}}}
440:{\displaystyle {\mathcal {A}}}
395:
374:{\displaystyle {\mathcal {A}}}
350:{\displaystyle {\mathcal {C}}}
323:{\displaystyle {\mathcal {C}}}
299:{\displaystyle {\mathcal {C}}}
225:{\displaystyle {\mathcal {C}}}
181:{\displaystyle {\mathcal {A}}}
148:
137:
131:
120:
92:{\displaystyle {\mathcal {C}}}
61:{\displaystyle {\mathcal {A}}}
1:
932:is localizing if and only if
908:), then a Serre subcategory
232:if and only if the objects
1200:
1035:and the quotient category
24:form important classes of
861:. The section functor is
872:If the abelian category
659:if the quotient functor
603:Localizing subcategories
22:localizing subcategories
1076:Gabriel-Popescu theorem
333:Each Serre subcategory
1127:
1107:
1065:
1029:
1005:
981:
950:
926:
890:
851:
827:
804:
780:
760:
705:
649:
625:
593:
569:
517:
493:
441:
409:
375:
351:
324:
300:
276:
251:
226:
202:
182:
158:
107:), if for every short
93:
62:
1128:
1108:
1066:
1030:
1006:
982:
951:
927:
906:Grothendieck category
891:
852:
833:) is also called the
828:
805:
781:
761:
706:
650:
626:
594:
570:
518:
494:
442:
410:
376:
352:
325:
301:
277:
252:
227:
203:
183:
159:
94:
63:
1117:
1085:
1039:
1015:
991:
967:
936:
912:
876:
841:
835:localization functor
814:
794:
770:
718:
663:
635:
611:
579:
527:
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385:
361:
337:
310:
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236:
212:
192:
168:
114:
79:
48:
1184:Homological algebra
275:{\displaystyle A''}
72:. A non-empty full
40:Serre subcategories
1147:Giraud subcategory
1123:
1103:
1061:
1025:
1001:
977:
946:
922:
886:
847:
826:{\displaystyle ST}
823:
800:
776:
756:
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645:
621:
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437:
405:
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250:{\displaystyle A'}
247:
222:
198:
178:
154:
89:
58:
1126:{\displaystyle R}
904:(e.g. if it is a
850:{\displaystyle S}
803:{\displaystyle T}
779:{\displaystyle T}
451:(in the sense of
449:quotient category
201:{\displaystyle A}
105:dense subcategory
101:Serre subcategory
34:quotient category
1191:
1164:
1132:
1130:
1129:
1124:
1112:
1110:
1109:
1104:
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1053:
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777:
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762:
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738:
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710:
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654:
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646:
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643:
630:
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627:
622:
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619:
598:
596:
595:
590:
588:
587:
575:whose kernel is
574:
572:
571:
566:
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563:
557:
552:
551:
542:
541:
522:
520:
519:
514:
512:
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498:
496:
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490:
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487:
481:
476:
475:
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438:
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414:
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394:
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372:
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369:
356:
354:
353:
348:
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329:
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321:
319:
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305:
303:
302:
297:
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281:
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256:
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231:
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187:
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163:
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130:
98:
96:
95:
90:
88:
87:
70:abelian category
67:
65:
64:
59:
57:
56:
30:abelian category
16:In mathematics,
1199:
1198:
1194:
1193:
1192:
1190:
1189:
1188:
1179:Category theory
1169:
1168:
1162:
1159:Nicolae Popescu
1155:
1143:
1115:
1114:
1083:
1082:
1080:module category
1037:
1036:
1013:
1012:
989:
988:
965:
964:
934:
933:
910:
909:
902:injective hulls
874:
873:
859:section functor
839:
838:
812:
811:
792:
791:
768:
767:
716:
715:
661:
660:
633:
632:
609:
608:
605:
577:
576:
525:
524:
501:
500:
465:
464:
425:
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383:
382:
359:
358:
335:
334:
308:
307:
284:
283:
264:
259:
258:
239:
234:
233:
210:
209:
190:
189:
166:
165:
140:
123:
112:
111:
77:
76:
46:
45:
42:
12:
11:
5:
1197:
1195:
1187:
1186:
1181:
1171:
1170:
1167:
1166:
1154:
1151:
1150:
1149:
1142:
1139:
1122:
1102:
1099:
1096:
1093:
1090:
1058:
1052:
1046:
1022:
998:
974:
943:
919:
883:
867:fully faithful
846:
822:
819:
810:(or sometimes
799:
775:
753:
748:
743:
737:
731:
726:
723:
698:
692:
686:
681:
676:
671:
668:
642:
618:
604:
601:
586:
562:
556:
550:
545:
540:
535:
532:
510:
486:
480:
474:
434:
421:exact functors
402:
397:
392:
368:
344:
317:
293:
270:
267:
245:
242:
219:
197:
175:
153:
150:
146:
143:
139:
136:
133:
129:
126:
122:
119:
109:exact sequence
86:
55:
41:
38:
13:
10:
9:
6:
4:
3:
2:
1196:
1185:
1182:
1180:
1177:
1176:
1174:
1160:
1157:
1156:
1152:
1148:
1145:
1144:
1140:
1138:
1136:
1120:
1097:
1091:
1088:
1081:
1077:
1072:
1050:
961:
959:
958:torsion class
907:
903:
899:
870:
868:
864:
860:
844:
836:
820:
817:
797:
789:
773:
766:. Since then
735:
724:
721:
714:
713:right adjoint
690:
669:
666:
658:
602:
600:
554:
533:
530:
478:
462:
458:
454:
450:
422:
418:
331:
268:
265:
243:
240:
195:
151:
144:
141:
134:
127:
124:
117:
110:
106:
102:
75:
71:
39:
37:
35:
31:
27:
26:subcategories
23:
19:
1073:
962:
896:is moreover
871:
858:
834:
656:
606:
457:Grothendieck
332:
306:. In words:
104:
100:
99:is called a
43:
21:
17:
15:
1133:a suitable
188:the object
103:(or also a
74:subcategory
1173:Categories
1153:References
898:cocomplete
863:left-exact
657:localizing
655:is called
282:belong to
1092:
747:→
725::
680:→
670::
544:→
534::
396:→
149:→
138:→
132:→
121:→
1161:; 1973;
1141:See also
900:and has
788:colimits
269:″
244:′
145:″
128:′
453:Gabriel
1113:(with
837:, and
711:has a
447:) the
208:is in
68:be an
28:of an
461:Serre
417:exact
18:Serre
1135:ring
1074:The
865:and
857:the
607:Let
257:and
44:Let
20:and
1089:Mod
963:If
415:is
357:of
164:in
1175::
960:.
869:.
599:.
463:)
459:,
455:,
36:.
1121:R
1101:)
1098:R
1095:(
1057:C
1051:/
1045:A
1021:C
997:C
973:A
942:C
918:C
882:A
845:S
821:T
818:S
798:T
774:T
752:A
742:C
736:/
730:A
722:S
697:C
691:/
685:A
675:A
667:T
641:C
617:A
585:C
561:C
555:/
549:A
539:A
531:T
509:A
485:C
479:/
473:A
433:A
401:A
391:C
367:A
343:C
316:C
292:C
266:A
241:A
218:C
196:A
174:A
152:0
142:A
135:A
125:A
118:0
85:C
54:A
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