1325:
1227:
In both homological and cohomological case there are also low degree exact sequences for spectral sequences in the third quadrant. When additional terms of the spectral sequence are known to vanish, the exact sequences can sometimes be extended further. For example, the
1222:
519:). Because the spectral sequence lies in the first quadrant, the degree one graded piece is equal to the first subgroup in the filtration defining the graded pieces. The inclusion of this subgroup yields the injection
1006:
1069:
99:
255:
1077:
381:
140:
282:
511:. Therefore the (1,0) term of the spectral sequence has converged, meaning that it is isomorphic to the degree one graded piece of the abutment
1295:
1258:
294:
685:, the last term in the short exact sequence can be replaced with the differential. This produces a four-term exact sequence. The map
467:
The sequence is a consequence of the definition of convergence of a spectral sequence. The second page differential with codomain
918:
745:, again because the spectral sequence lies in the first quadrant, and hence the spectral sequence has converged. Consequently
590:. At the third page, the (0, 1) term of the spectral sequence has converged, because all the differentials into and out of
1346:
1014:
44:
198:
548:
term of the spectral sequence has not converged. It has a potentially non-trivial differential leading to
805:
The five-term exact sequence can be extended at the cost of making one of the terms less explicit. The
1324:
1217:{\displaystyle H_{2}(A)\to E_{2,0}^{2}\xrightarrow {d_{2}} E_{0,1}^{2}\to H_{1}(A)\to E_{1,0}^{2}\to 0}
1229:
1330:
449:
107:
1242:
1309:
1291:
1254:
32:
627:) by the first subgroup in the filtration, and hence it is the cokernel of the edge map from
1272:
377:
1305:
1268:
495:, which is also zero by assumption. Similarly, the incoming and outgoing differentials of
260:
1301:
1276:
1264:
441:
438:
430:
790:), which is another edge map, therefore has kernel equal to the differential landing at
1290:. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press.
1283:
28:
1340:
1232:
associated to a short exact sequence of complexes can be derived in this manner.
1320:
1313:
534:) which begins the five-term exact sequence. This injection is called an
898:), its kernel is not the previous term in the seven-term exact sequence.
452:
1128:
16:
A sequence of terms related to the first step of a spectral sequence
481:, which is zero by assumption. The differential with domain
1001:{\displaystyle 0\to H^{0}(A)\to E_{1}^{0,0}\to E_{1}^{0,1}.}
380:
arises as the five-term exact sequence associated to the
875:
This sequence does not immediately extend with a map to
150:
are both non-negative. Then there is an exact sequence
901:
For spectral sequences whose first interesting page is
1080:
1017:
921:
263:
201:
110:
47:
797:. This completes the construction of the sequence.
597:
either begin or end outside the first quadrant when
104:
be a first quadrant spectral sequence, meaning that
1216:
1063:
1000:
276:
249:
134:
93:
752:is isomorphic to the degree two graded piece of
1064:{\displaystyle E_{p,q}^{2}\Rightarrow H_{n}(A)}
94:{\displaystyle E_{2}^{p,q}\Rightarrow H^{n}(A)}
1011:Similarly for a homological spectral sequence
731:. The incoming and outgoing differentials of
1251:Grundlehren der Mathematischen Wissenschaften
8:
1245:; Schmidt, Alexander; Wingberg, Kay (2000),
912:analogous to the five-term exact sequence:
1253:, vol. 323, Berlin: Springer-Verlag,
250:{\displaystyle E_{2}^{0,1}\to E_{2}^{2,0}}
1202:
1191:
1169:
1156:
1145:
1133:
1118:
1107:
1085:
1079:
1046:
1033:
1022:
1016:
983:
978:
959:
954:
932:
920:
619:). This graded piece is the quotient of
382:Lyndon–Hochschild–Serre spectral sequence
268:
262:
235:
230:
211:
206:
200:
120:
115:
109:
76:
57:
52:
46:
555:. However, the differential landing at
31:of terms related to the first step of a
760:). In particular, it is a subgroup of
1288:An introduction to homological algebra
634:. This yields a short exact sequence
7:
717:is the cokernel of the differential
295:inflation-restriction exact sequence
611:is the degree zero graded piece of
671:is the kernel of the differential
576:is the kernel of the differential
25:exact sequence of low-degree terms
14:
1323:
284:-term of the spectral sequence.
883:). While there is an edge map
569:, which is zero, and therefore
1208:
1184:
1181:
1175:
1162:
1100:
1097:
1091:
1058:
1052:
1039:
971:
947:
944:
938:
925:
223:
88:
82:
69:
1:
703:The outgoing differential of
700:is also called an edge map.
1247:Cohomology of Number Fields
257:is the differential of the
135:{\displaystyle E_{2}^{p,q}}
1363:
1071:we get an exact sequence:
910:three-term exact sequence
807:seven-term exact sequence
21:five-term exact sequence
1218:
1065:
1002:
278:
251:
136:
95:
1219:
1066:
1003:
279:
277:{\displaystyle E_{2}}
252:
142:vanishes except when
137:
96:
1078:
1015:
919:
261:
199:
108:
45:
38:More precisely, let
1230:long exact sequence
1207:
1161:
1139:
1123:
1038:
994:
970:
246:
222:
131:
68:
1347:Spectral sequences
1331:Mathematics portal
1284:Weibel, Charles A.
1214:
1187:
1141:
1103:
1061:
1018:
998:
974:
950:
768:). The composite
274:
247:
226:
202:
132:
111:
91:
48:
1297:978-0-521-55987-4
1260:978-3-540-66671-4
1140:
504:are zero for all
33:spectral sequence
1354:
1333:
1328:
1327:
1317:
1279:
1243:Neukirch, JĂĽrgen
1223:
1221:
1220:
1215:
1206:
1201:
1174:
1173:
1160:
1155:
1138:
1137:
1124:
1122:
1117:
1090:
1089:
1070:
1068:
1067:
1062:
1051:
1050:
1037:
1032:
1007:
1005:
1004:
999:
993:
982:
969:
958:
937:
936:
744:
604:. Consequently
603:
510:
474:originates from
378:group cohomology
283:
281:
280:
275:
273:
272:
256:
254:
253:
248:
245:
234:
221:
210:
141:
139:
138:
133:
130:
119:
100:
98:
97:
92:
81:
80:
67:
56:
19:In mathematics,
1362:
1361:
1357:
1356:
1355:
1353:
1352:
1351:
1337:
1336:
1329:
1322:
1298:
1282:
1261:
1241:
1238:
1165:
1129:
1081:
1076:
1075:
1042:
1013:
1012:
928:
917:
916:
907:
889:
870:
863:
856:
841:
834:
819:
803:
796:
781:
774:
751:
739:
737:
730:
723:
716:
709:
699:
684:
677:
670:
659:
644:
633:
610:
598:
596:
589:
582:
575:
568:
561:
554:
547:
525:
505:
503:
494:
487:
480:
473:
465:
442:normal subgroup
431:profinite group
290:
264:
259:
258:
197:
196:
182:
175:
160:
106:
105:
72:
43:
42:
17:
12:
11:
5:
1360:
1358:
1350:
1349:
1339:
1338:
1335:
1334:
1319:
1318:
1296:
1280:
1259:
1237:
1234:
1225:
1224:
1213:
1210:
1205:
1200:
1197:
1194:
1190:
1186:
1183:
1180:
1177:
1172:
1168:
1164:
1159:
1154:
1151:
1148:
1144:
1136:
1132:
1127:
1121:
1116:
1113:
1110:
1106:
1102:
1099:
1096:
1093:
1088:
1084:
1060:
1057:
1054:
1049:
1045:
1041:
1036:
1031:
1028:
1025:
1021:
1009:
1008:
997:
992:
989:
986:
981:
977:
973:
968:
965:
962:
957:
953:
949:
946:
943:
940:
935:
931:
927:
924:
905:
887:
873:
872:
868:
861:
854:
839:
832:
817:
802:
799:
794:
779:
772:
749:
735:
728:
721:
714:
707:
697:
682:
675:
668:
662:
661:
657:
642:
631:
608:
594:
587:
580:
573:
566:
559:
552:
545:
523:
499:
492:
485:
478:
471:
464:
461:
460:
459:
423:
422:
421:
374:
373:
372:
298:
297:
289:
286:
271:
267:
244:
241:
238:
233:
229:
225:
220:
217:
214:
209:
205:
195:Here, the map
193:
192:
180:
173:
158:
129:
126:
123:
118:
114:
102:
101:
90:
87:
84:
79:
75:
71:
66:
63:
60:
55:
51:
15:
13:
10:
9:
6:
4:
3:
2:
1359:
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1342:
1332:
1326:
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1315:
1311:
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1303:
1299:
1293:
1289:
1285:
1281:
1278:
1274:
1270:
1266:
1262:
1256:
1252:
1248:
1244:
1240:
1239:
1235:
1233:
1231:
1211:
1203:
1198:
1195:
1192:
1188:
1178:
1170:
1166:
1157:
1152:
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1146:
1142:
1134:
1130:
1125:
1119:
1114:
1111:
1108:
1104:
1094:
1086:
1082:
1074:
1073:
1072:
1055:
1047:
1043:
1034:
1029:
1026:
1023:
1019:
995:
990:
987:
984:
979:
975:
966:
963:
960:
955:
951:
941:
933:
929:
922:
915:
914:
913:
911:
908:, there is a
904:
899:
897:
893:
886:
882:
878:
867:
860:
853:
849:
845:
838:
831:
827:
823:
816:
812:
811:
810:
808:
800:
798:
793:
789:
785:
778:
771:
767:
763:
759:
755:
748:
742:
734:
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720:
713:
706:
701:
696:
692:
688:
681:
674:
667:
656:
652:
648:
641:
637:
636:
635:
630:
626:
622:
618:
614:
607:
601:
593:
586:
579:
572:
565:
558:
551:
544:
539:
537:
533:
529:
522:
518:
514:
508:
502:
498:
491:
488:has codomain
484:
477:
470:
462:
457:
455:
451:
447:
443:
440:
436:
432:
428:
424:
420:
416:
412:
408:
404:
400:
396:
392:
388:
385:
384:
383:
379:
375:
370:
366:
362:
358:
354:
350:
346:
342:
338:
334:
330:
326:
322:
318:
314:
310:
306:
302:
301:
300:
299:
296:
292:
291:
287:
285:
269:
265:
242:
239:
236:
231:
227:
218:
215:
212:
207:
203:
190:
186:
179:
172:
168:
164:
157:
153:
152:
151:
149:
145:
127:
124:
121:
116:
112:
85:
77:
73:
64:
61:
58:
53:
49:
41:
40:
39:
36:
34:
30:
26:
22:
1287:
1250:
1246:
1226:
1010:
909:
902:
900:
895:
891:
884:
880:
876:
874:
865:
858:
851:
847:
843:
836:
829:
825:
821:
814:
806:
804:
791:
787:
783:
776:
769:
765:
761:
757:
753:
746:
740:
738:are zero if
732:
725:
718:
711:
710:is zero, so
704:
702:
694:
690:
686:
679:
672:
665:
663:
654:
650:
646:
639:
628:
624:
620:
616:
612:
605:
599:
591:
584:
577:
570:
563:
556:
549:
542:
540:
535:
531:
527:
520:
516:
512:
506:
500:
496:
489:
482:
475:
468:
466:
463:Construction
453:
445:
434:
426:
418:
414:
410:
406:
402:
398:
394:
390:
386:
368:
364:
360:
356:
352:
348:
344:
340:
336:
332:
328:
324:
320:
316:
312:
308:
304:
194:
188:
184:
177:
170:
166:
162:
155:
147:
143:
103:
37:
24:
20:
18:
1277:0948.11001
1236:References
801:Variations
562:begins at
1209:→
1185:→
1163:→
1101:→
1040:⇒
972:→
948:→
926:→
224:→
70:⇒
1341:Category
1314:36131259
1286:(1994).
1126:→
664:Because
536:edge map
450:discrete
29:sequence
1306:1269324
1269:1737196
456:-module
288:Example
1312:
1304:
1294:
1275:
1267:
1257:
842:→ Ker(
444:, and
439:closed
425:where
448:is a
437:is a
429:is a
409:)) ⇒
27:is a
1310:OCLC
1292:ISBN
1255:ISBN
857:) →
850:) →
828:) →
813:0 →
693:) →
660:→ 0.
653:) →
638:0 →
541:The
343:) →
331:) →
319:) →
303:0 →
293:The
169:) →
154:0 →
146:and
1273:Zbl
809:is
743:≥ 3
602:≥ 3
509:≥ 2
415:G,
376:in
359:) →
23:or
1343::
1308:.
1302:MR
1300:.
1271:,
1265:MR
1263:,
1249:,
890:→
864:→
835:→
820:→
782:→
775:→
724:→
678:→
645:→
583:→
538:.
526:→
433:,
405:,
397:,
367:,
355:,
339:,
327:,
315:,
191:).
183:→
176:→
161:→
35:.
1316:.
1212:0
1204:2
1199:0
1196:,
1193:1
1189:E
1182:)
1179:A
1176:(
1171:1
1167:H
1158:2
1153:1
1150:,
1147:0
1143:E
1135:2
1131:d
1120:2
1115:0
1112:,
1109:2
1105:E
1098:)
1095:A
1092:(
1087:2
1083:H
1059:)
1056:A
1053:(
1048:n
1044:H
1035:2
1030:q
1027:,
1024:p
1020:E
996:.
991:1
988:,
985:0
980:1
976:E
967:0
964:,
961:0
956:1
952:E
945:)
942:A
939:(
934:0
930:H
923:0
906:1
903:E
896:A
894:(
892:H
888:2
885:E
881:A
879:(
877:H
871:.
869:2
866:E
862:2
859:E
855:2
852:E
848:A
846:(
844:H
840:2
837:E
833:2
830:E
826:A
824:(
822:H
818:2
815:E
795:2
792:E
788:A
786:(
784:H
780:3
777:E
773:2
770:E
766:A
764:(
762:H
758:A
756:(
754:H
750:3
747:E
741:r
736:r
733:E
729:2
726:E
722:2
719:E
715:3
712:E
708:2
705:E
698:2
695:E
691:A
689:(
687:H
683:2
680:E
676:2
673:E
669:3
666:E
658:3
655:E
651:A
649:(
647:H
643:2
640:E
632:2
629:E
625:A
623:(
621:H
617:A
615:(
613:H
609:3
606:E
600:r
595:r
592:E
588:2
585:E
581:2
578:E
574:3
571:E
567:2
564:E
560:2
557:E
553:2
550:E
546:2
543:E
532:A
530:(
528:H
524:2
521:E
517:A
515:(
513:H
507:r
501:r
497:E
493:2
490:E
486:2
483:E
479:2
476:E
472:2
469:E
458:.
454:G
446:A
435:N
427:G
419:)
417:A
413:(
411:H
407:A
403:N
401:(
399:H
395:N
393:/
391:G
389:(
387:H
371:)
369:A
365:G
363:(
361:H
357:A
353:N
351:/
349:G
347:(
345:H
341:A
337:N
335:(
333:H
329:A
325:G
323:(
321:H
317:A
313:N
311:/
309:G
307:(
305:H
270:2
266:E
243:0
240:,
237:2
232:2
228:E
219:1
216:,
213:0
208:2
204:E
189:A
187:(
185:H
181:2
178:E
174:2
171:E
167:A
165:(
163:H
159:2
156:E
148:q
144:p
128:q
125:,
122:p
117:2
113:E
89:)
86:A
83:(
78:n
74:H
65:q
62:,
59:p
54:2
50:E
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