51:
1414:
376:
154:
253:
668:
1201:
1027:
1251:
801:
458:
857:
712:
917:
827:
1122:
979:
554:
522:
490:
587:
112:
1142:
1087:
1067:
1047:
414:
273:
194:
174:
1263:
1479:
1455:
1364:
383:
981:
be a
Waldhausen category that has a cylinder functor satisfying the Cylinder Axiom. Suppose there is a surjective homomorphism
278:
117:
35:
1474:
1448:
595:
199:
1484:
1147:
865:
1441:
984:
674:
1206:
748:
557:
419:
1391:
20:
836:
691:
896:
1360:
936:
806:
1352:
1092:
952:
527:
495:
463:
34:
is a subcategory of another exact category, we mean it is strictly full subcategory (i.e.,
563:
91:
1425:
1127:
1072:
1052:
1032:
399:
258:
179:
159:
31:
1380:
416:
be the category with cofibrations, equipped with two categories of weak equivalences,
50:
1468:
1374:
114:
be exact categories (or other variants). Given a short exact sequence of functors
1336:
1413:
1386:
Harris, Tom (2013). "Algebraic proofs of some fundamental theorems in algebraic
928:
1421:
1356:
1396:
45:
371:{\displaystyle F_{*}=F'_{*}+F''_{*}:K_{i}(B)\to K_{i}(C)}
149:{\displaystyle F'\rightarrowtail F\twoheadrightarrow F''}
1429:
62:
1209:
1150:
1130:
1095:
1075:
1055:
1035:
987:
955:
899:
839:
809:
751:
694:
598:
566:
530:
498:
466:
422:
402:
281:
261:
202:
182:
162:
120:
94:
589:
satisfies the
Saturation and Extension Axioms. Then
19:
In mathematics, there are several theorems basic to
737:admits a resolution of finite length by objects in
726:and under the kernels of admissible surjections in
1245:
1195:
1136:
1116:
1081:
1061:
1041:
1021:
973:
911:
851:
821:
795:
706:
662:
581:
548:
516:
484:
452:
408:
370:
267:
247:
188:
168:
148:
106:
1049:denote the full Waldhausen subcategory of all
30:Throughout, for simplicity, we assume when an
1449:
1375:On Fundamental Theorems of Algebraic K-Theory
1341:-book: An introduction to algebraic K-theory"
1297:, Ch. V, Waldhausen Localization Theorem 2.1.
8:
663:{\displaystyle K(A^{w})\to K(A,v)\to K(A,w)}
943:
682:
390:
384:localization theorem for abelian categories
82:
1456:
1442:
1381:FUNDAMENTAL THEOREMS OF ALGEBRAIC K-THEORY
248:{\displaystyle F_{*}\simeq F'_{*}+F''_{*}}
1395:
1264:Fundamental theorem of algebraic K-theory
1208:
1149:
1129:
1094:
1074:
1054:
1034:
998:
986:
954:
898:
838:
808:
778:
756:
750:
693:
609:
597:
565:
529:
497:
465:
421:
401:
382:The localization theorem generalizes the
353:
331:
315:
299:
286:
280:
260:
236:
220:
207:
201:
181:
161:
119:
93:
560:satisfying the Cylinder Axiom, and that
1275:
873:if (i) it is closed under extension in
524:are both Waldhausen categories. Assume
1318:
1306:
1294:
1282:
7:
1410:
1408:
1196:{\displaystyle v.s.B\to v.s.A\to BG}
1347:. Graduate Studies in Mathematics.
923:. The prototypical example is when
1022:{\displaystyle \pi :K_{0}(A)\to G}
14:
1246:{\displaystyle K(B)\to K(A)\to G}
796:{\displaystyle K_{i}(C)=K_{i}(D)}
1412:
1321:, Ch. V, Cofinality Theorem 2.3.
1309:, Ch. V, Resolution Theorem 3.1.
1285:, Ch. V, Additivity Theorem 1.2.
453:{\displaystyle v(A)\subset w(A)}
49:
391:Waldhausen Localization Theorem
1480:Theorems in algebraic topology
1237:
1234:
1228:
1222:
1219:
1213:
1184:
1166:
1105:
1099:
1013:
1010:
1004:
968:
956:
790:
784:
768:
762:
722:is closed under extensions in
657:
645:
639:
636:
624:
618:
615:
602:
576:
570:
543:
531:
511:
499:
479:
467:
447:
441:
432:
426:
365:
359:
346:
343:
337:
135:
129:
1:
1428:. You can help Knowledge by
877:and if (ii) for each object
714:be exact categories. Assume
275:-space maps; consequently,
1501:
1407:
859:be exact categories. Then
852:{\displaystyle C\subset D}
707:{\displaystyle C\subset D}
16:Four mathematical theorems
1253:are homotopy fibrations.
912:{\displaystyle M\oplus N}
1345:Graduate Studies in Math
1335:Weibel, Charles (2013).
822:{\displaystyle i\geq 0}
1424:-related article is a
1247:
1197:
1138:
1118:
1117:{\displaystyle \pi =0}
1083:
1063:
1043:
1023:
975:
913:
853:
823:
797:
708:
664:
583:
550:
518:
486:
454:
410:
372:
269:
249:
190:
170:
150:
108:
1379:GABE ANGELINI-KNOLL,
1248:
1198:
1139:
1119:
1084:
1064:
1044:
1024:
976:
974:{\displaystyle (A,v)}
914:
854:
824:
798:
733:(ii) Every object in
709:
665:
584:
551:
549:{\displaystyle (A,w)}
519:
517:{\displaystyle (A,w)}
487:
485:{\displaystyle (A,v)}
455:
411:
373:
270:
250:
191:
171:
151:
109:
1207:
1148:
1128:
1093:
1073:
1053:
1033:
985:
953:
897:
837:
807:
749:
692:
596:
582:{\displaystyle w(A)}
564:
528:
496:
464:
420:
400:
279:
259:
200:
180:
160:
118:
92:
1373:Ross E. Staffeldt,
947: —
935:is the category of
927:is the category of
686: —
394: —
323:
307:
244:
228:
107:{\displaystyle B,C}
86: —
1475:Algebraic K-theory
1243:
1203:and its delooping
1193:
1134:
1114:
1079:
1059:
1039:
1019:
971:
945:
944:Cofinality theorem
937:projective modules
909:
849:
819:
793:
704:
684:
683:Resolution theorem
675:homotopy fibration
660:
579:
546:
514:
482:
450:
406:
392:
368:
311:
295:
265:
245:
232:
216:
186:
166:
146:
104:
84:
83:Additivity theorem
61:. You can help by
36:isomorphism-closed
1437:
1436:
1366:978-0-8218-9132-2
1137:{\displaystyle G}
1082:{\displaystyle A}
1062:{\displaystyle X}
1042:{\displaystyle B}
409:{\displaystyle A}
268:{\displaystyle H}
189:{\displaystyle C}
169:{\displaystyle B}
79:
78:
1492:
1458:
1451:
1444:
1416:
1409:
1401:
1399:
1370:
1322:
1316:
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1194:
1143:
1141:
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1123:
1121:
1120:
1115:
1088:
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1080:
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1048:
1046:
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1028:
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1003:
1002:
980:
978:
977:
972:
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918:
916:
915:
910:
858:
856:
855:
850:
828:
826:
825:
820:
802:
800:
799:
794:
783:
782:
761:
760:
713:
711:
710:
705:
687:
669:
667:
666:
661:
614:
613:
588:
586:
585:
580:
558:cylinder functor
555:
553:
552:
547:
523:
521:
520:
515:
491:
489:
488:
483:
459:
457:
456:
451:
415:
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175:
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145:
128:
113:
111:
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87:
74:
71:
53:
46:
1500:
1499:
1495:
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1490:
1489:
1465:
1464:
1463:
1462:
1405:
1385:
1367:
1357:10.1090/gsm/145
1334:
1331:
1326:
1325:
1317:
1313:
1305:
1301:
1293:
1289:
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1255:
1205:
1204:
1146:
1145:
1126:
1125:
1091:
1090:
1071:
1070:
1051:
1050:
1031:
1030:
994:
983:
982:
951:
950:
946:
895:
894:
835:
834:
831:
805:
804:
774:
752:
747:
746:
690:
689:
685:
679:
605:
594:
593:
562:
561:
526:
525:
494:
493:
462:
461:
418:
417:
398:
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380:
349:
327:
282:
277:
276:
257:
256:
203:
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197:
178:
177:
158:
157:
138:
121:
116:
115:
90:
89:
85:
75:
69:
66:
59:needs expansion
44:
17:
12:
11:
5:
1498:
1496:
1488:
1487:
1482:
1477:
1467:
1466:
1461:
1460:
1453:
1446:
1438:
1435:
1434:
1417:
1403:
1402:
1383:
1377:
1371:
1365:
1330:
1327:
1324:
1323:
1311:
1299:
1287:
1274:
1273:
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1268:
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1266:
1259:
1256:
1242:
1239:
1236:
1233:
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1227:
1224:
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1218:
1215:
1212:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1133:
1113:
1110:
1107:
1104:
1101:
1098:
1078:
1058:
1038:
1018:
1015:
1012:
1009:
1006:
1001:
997:
993:
990:
970:
967:
964:
961:
958:
941:
908:
905:
902:
863:is said to be
848:
845:
842:
818:
815:
812:
792:
789:
786:
781:
777:
773:
770:
767:
764:
759:
755:
743:
742:
731:
703:
700:
697:
680:
671:
670:
659:
656:
653:
650:
647:
644:
641:
638:
635:
632:
629:
626:
623:
620:
617:
612:
608:
604:
601:
578:
575:
572:
569:
545:
542:
539:
536:
533:
513:
510:
507:
504:
501:
481:
478:
475:
472:
469:
449:
446:
443:
440:
437:
434:
431:
428:
425:
405:
388:
367:
364:
361:
356:
352:
348:
345:
342:
339:
334:
330:
326:
322:
318:
314:
310:
306:
302:
298:
294:
289:
285:
264:
243:
239:
235:
231:
227:
223:
219:
215:
210:
206:
185:
165:
144:
141:
137:
134:
131:
127:
124:
103:
100:
97:
80:
77:
76:
56:
54:
43:
40:
32:exact category
15:
13:
10:
9:
6:
4:
3:
2:
1497:
1486:
1485:Algebra stubs
1483:
1481:
1478:
1476:
1473:
1472:
1470:
1459:
1454:
1452:
1447:
1445:
1440:
1439:
1433:
1431:
1427:
1423:
1418:
1415:
1411:
1406:
1398:
1393:
1389:
1384:
1382:
1378:
1376:
1372:
1368:
1362:
1358:
1354:
1350:
1346:
1342:
1340:
1333:
1332:
1328:
1320:
1315:
1312:
1308:
1303:
1300:
1296:
1291:
1288:
1284:
1279:
1276:
1269:
1265:
1262:
1261:
1257:
1254:
1240:
1231:
1225:
1216:
1210:
1190:
1187:
1181:
1178:
1175:
1172:
1169:
1163:
1160:
1157:
1154:
1151:
1131:
1111:
1108:
1102:
1096:
1076:
1056:
1036:
1016:
1007:
999:
995:
991:
988:
965:
962:
959:
940:
938:
934:
930:
926:
922:
906:
903:
900:
892:
888:
884:
880:
876:
872:
868:
867:
862:
846:
843:
840:
830:
816:
813:
810:
787:
779:
775:
771:
765:
757:
753:
740:
736:
732:
729:
725:
721:
717:
716:
715:
701:
698:
695:
678:
676:
654:
651:
648:
642:
633:
630:
627:
621:
610:
606:
599:
592:
591:
590:
573:
567:
559:
540:
537:
534:
508:
505:
502:
476:
473:
470:
444:
438:
435:
429:
423:
403:
387:
385:
379:
362:
354:
350:
340:
332:
328:
324:
320:
316:
312:
308:
304:
300:
296:
292:
287:
283:
262:
241:
237:
233:
229:
225:
221:
217:
213:
208:
204:
183:
163:
142:
139:
132:
125:
122:
101:
98:
95:
73:
64:
60:
57:This section
55:
52:
48:
47:
41:
39:
37:
33:
28:
26:
24:
1430:expanding it
1419:
1404:
1387:
1348:
1344:
1338:
1329:Bibliography
1314:
1302:
1290:
1278:
942:
932:
929:free modules
924:
920:
890:
886:
885:there is an
882:
878:
874:
870:
864:
860:
832:
744:
738:
734:
727:
723:
719:
681:
672:
460:, such that
389:
381:
81:
70:October 2019
67:
63:adding to it
58:
29:
22:
18:
1319:Weibel 2013
1307:Weibel 2013
1295:Weibel 2013
1283:Weibel 2013
1469:Categories
1390:-theory".
1270:References
893:such that
21:algebraic
1397:1311.5162
1238:→
1223:→
1185:→
1167:→
1097:π
1014:→
989:π
904:⊕
844:⊂
814:≥
699:⊂
640:→
619:→
436:⊂
347:→
317:∗
301:∗
288:∗
238:∗
222:∗
214:≃
209:∗
136:↠
130:↣
1258:See also
1029:and let
803:for all
321:″
305:′
242:″
226:′
143:″
126:′
42:Theorems
1422:algebra
1144:. Then
866:cofinal
25:-theory
1363:
919:is in
556:has a
1420:This
1392:arXiv
1337:"The
1089:with
745:Then
673:is a
156:from
1426:stub
1361:ISBN
949:Let
931:and
833:Let
718:(i)
688:Let
492:and
396:Let
88:Let
1353:doi
1349:145
1124:in
1069:in
889:in
881:in
869:in
255:as
176:to
65:.
38:.)
1471::
1359:.
1351:.
1343:.
939:.
829:.
677:.
386:.
378:.
196:,
27:.
1457:e
1450:t
1443:v
1432:.
1400:.
1394::
1388:K
1369:.
1355::
1339:K
1241:G
1235:)
1232:A
1229:(
1226:K
1220:)
1217:B
1214:(
1211:K
1191:G
1188:B
1182:A
1179:.
1176:s
1173:.
1170:v
1164:B
1161:.
1158:s
1155:.
1152:v
1132:G
1112:0
1109:=
1106:]
1103:X
1100:[
1077:A
1057:X
1037:B
1017:G
1011:)
1008:A
1005:(
1000:0
996:K
992::
969:)
966:v
963:,
960:A
957:(
933:D
925:C
921:C
907:N
901:M
891:D
887:N
883:D
879:M
875:D
871:D
861:C
847:D
841:C
817:0
811:i
791:)
788:D
785:(
780:i
776:K
772:=
769:)
766:C
763:(
758:i
754:K
741:.
739:C
735:D
730:.
728:D
724:D
720:C
702:D
696:C
658:)
655:w
652:,
649:A
646:(
643:K
637:)
634:v
631:,
628:A
625:(
622:K
616:)
611:w
607:A
603:(
600:K
577:)
574:A
571:(
568:w
544:)
541:w
538:,
535:A
532:(
512:)
509:w
506:,
503:A
500:(
480:)
477:v
474:,
471:A
468:(
448:)
445:A
442:(
439:w
433:)
430:A
427:(
424:v
404:A
366:)
363:C
360:(
355:i
351:K
344:)
341:B
338:(
333:i
329:K
325::
313:F
309:+
297:F
293:=
284:F
263:H
234:F
230:+
218:F
205:F
184:C
164:B
140:F
133:F
123:F
102:C
99:,
96:B
72:)
68:(
23:K
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