908:. In this guise, all terms in the spectral sequence are of purely algebraic (as opposed to analytic) nature. In particular, the question of the degeneration of this spectral sequence makes sense for varieties over a field of characteristic
663:
806:
152:
375:
1129:
213:
477:
902:
871:
836:
507:
702:
1293:
553:
240:
1026:
over a field of characteristic 0, the Hodge–de Rham spectral sequence starting with
Hochschild homology and abutting to periodic cyclic homology, degenerates:
998:
The de Rham complex and also the de Rham cohomology of a variety admit generalizations to non-commutative geometry. This more general setup studies
564:
718:
72:
1147:
1159:
259:
30:
1032:
1523:
1533:
1528:
1007:
1291:
Frölicher, Alfred (1955), "Relations between the cohomology groups of
Dolbeault and topological invariants",
168:
712:
For smooth proper varieties over a field of characteristic 0, the spectral sequence can also be written as
839:
1142:
by adapting the above idea of
Deligne and Illusie to the generality of smooth and proper dg-categories.
389:
1396:
Kaledin, D. (2008), "Non-commutative Hodge-to-de Rham degeneration via the method of
Deligne-Illusie",
38:, who actually discovered it). This spectral sequence describes the precise relationship between the
1487:
1457:
1369:
243:
39:
1003:
979:
905:
556:
47:
35:
1503:
1477:
1447:
1431:
1405:
1385:
1312:
1270:
1231:
880:
849:
814:
525:
485:
518:
1338:
1223:
675:
250:
246:
63:
1468:
Mathew, Akhil (2020) , "Kaledin's degeneration theorem and topological
Hochschild homology",
380:
together with the usual spectral sequence resulting from a filtered object, in this case the
1495:
1415:
1377:
1328:
1302:
1262:
1215:
162:
43:
1427:
1324:
982:, which only exists in positive characteristic. This degeneration result in characteristic
531:
218:
1423:
1320:
1022:. Kontsevich and Soibelman conjectured in 2009 that for any smooth and proper dg category
1011:
874:
51:
25:
1491:
1461:
1373:
1349:
1169:
21:
1333:
1517:
1507:
1389:
1274:
1235:
1203:
1018:, these invariants give back differential forms, respectively, de Rham cohomology of
923:
672:. An extension of this degeneration in a relative situation, for a proper smooth map
1435:
1250:
986:>0 can then be used to also prove the degeneration for the spectral sequence for
1164:
669:
215:
is its cohomology with complex coefficients and the left hand term, which is the
46:. On a compact Kähler manifold, the sequence degenerates, thereby leading to the
1353:
999:
946:
1419:
242:-page of the spectral sequence, is the cohomology with values in the sheaf of
1227:
658:{\displaystyle \bigoplus _{p+q=n}H^{p}(X,\Omega ^{q})=H^{n}(X,\mathbf {C} ).}
1499:
517:
The central theorem related to this spectral sequence is that for a compact
1342:
1307:
1251:"Théorème de Lefschetz et Critères de Dégénérescence de Suites Spectrales"
1204:"Théorème de Lefschetz et Critères de Dégénérescence de Suites Spectrales"
249:. The existence of the spectral sequence as stated above follows from the
1208:
Publications Mathématiques de l'Institut des Hautes Études
Scientifiques
1381:
1266:
1219:
801:{\displaystyle H^{q}(X,\Omega ^{p})\Rightarrow H^{p+q}(X,\Omega ^{*}),}
147:{\displaystyle H^{q}(X,\Omega ^{p})\Rightarrow H^{p+q}(X,\mathbf {C} )}
1410:
1316:
1482:
1452:
838:
denotes the sheaf of algebraic differential forms (also known as
555:-page. In particular, it gives an isomorphism referred to as the
668:
The degeneration of the spectral sequence can be shown using
253:, which gives a quasi-isomorphism of complexes of sheaves
28:) is an alternative term sometimes used to describe the
1172:- useful for computing cohomology of Hodge decomposition
933:, the spectral sequence degenerates, provided that dim(
370:{\displaystyle \mathbf {C} \rightarrow \Omega ^{*}:=,}
1035:
883:
852:
817:
721:
678:
567:
534:
488:
392:
262:
221:
171:
75:
1124:{\displaystyle HH_{*}(C/k)\Rightarrow HP_{*}(C/k).}
1123:
896:
865:
830:
800:
696:
657:
547:
501:
471:
369:
234:
207:
146:
528:, the above spectral sequence degenerates at the
1294:Proceedings of the National Academy of Sciences
915:
1146:has given a proof of this degeneration using
945:admits a smooth proper lift over the ring of
8:
1360:et décomposition du complexe de de Rham",
1002:. To a dg category, one can associate its
1481:
1451:
1409:
1332:
1306:
1107:
1095:
1073:
1055:
1043:
1034:
888:
882:
857:
851:
822:
816:
786:
761:
745:
726:
720:
677:
644:
629:
613:
594:
572:
566:
539:
533:
493:
487:
448:
435:
407:
397:
391:
349:
331:
326:
324:
323:
317:
305:
300:
298:
297:
291:
275:
263:
261:
226:
220:
197:
176:
170:
136:
115:
99:
80:
74:
42:and the de Rham cohomology of a general
1182:
1139:
1135:
918:showed that for a smooth proper scheme
208:{\displaystyle H^{p+q}(X,\mathbf {C} )}
1444:Spectral sequences for cyclic homology
1398:Pure and Applied Mathematics Quarterly
1143:
7:
58:Description of the spectral sequence
1189:See for example Griffiths, Harris
1010:. When applied to the category of
990:over a field of characteristic 0.
957:) of length two (for example, for
885:
854:
819:
783:
742:
610:
490:
472:{\displaystyle F^{p}\Omega ^{*}:=}
445:
432:
404:
346:
314:
288:
272:
96:
14:
1191:Principles of algebraic geometry
904:with the differential being the
645:
264:
198:
137:
1148:topological Hochschild homology
18:Hodge–de Rham spectral sequence
1134:This conjecture was proved by
1115:
1101:
1085:
1082:
1066:
1063:
1049:
792:
773:
754:
751:
732:
688:
649:
635:
619:
600:
466:
460:
441:
428:
422:
416:
361:
342:
327:
301:
284:
268:
202:
188:
141:
127:
108:
105:
86:
1:
704:, was also shown by Deligne.
1356:(1987), "Relèvements modulo
916:Deligne & Illusie (1987)
1160:Frölicher spectral sequence
1014:on a smooth proper variety
929:of positive characteristic
897:{\displaystyle \Omega ^{q}}
866:{\displaystyle \Omega ^{*}}
831:{\displaystyle \Omega ^{p}}
502:{\displaystyle \Omega ^{*}}
31:Frölicher spectral sequence
1550:
1420:10.4310/PAMQ.2008.v4.n3.a8
1442:Kaledin, Dmitry (2016),
1249:Deligne, Pierre (1968),
1008:periodic cyclic homology
978:). Their proof uses the
697:{\displaystyle f:X\to S}
1500:10.2140/gt.2020.24.2675
1470:Geometry & Topology
994:Non-commutative version
1125:
898:
867:
832:
802:
708:Purely algebraic proof
698:
659:
549:
503:
473:
371:
236:
209:
148:
1308:10.1073/pnas.41.9.641
1126:
970:, this ring would be
899:
868:
833:
803:
699:
660:
550:
548:{\displaystyle E_{1}}
504:
474:
372:
237:
235:{\displaystyle E_{1}}
210:
149:
1202:Deligne, P. (1968).
1033:
881:
877:, consisting of the
850:
840:Kähler differentials
815:
719:
676:
565:
532:
486:
390:
260:
219:
169:
73:
40:Dolbeault cohomology
20:(named in honor of
16:In mathematics, the
1524:Cohomology theories
1492:2017arXiv171009045M
1462:2016arXiv160100637K
1374:1987InMat..89..247D
1004:Hochschild homology
980:Cartier isomorphism
906:exterior derivative
873:is the (algebraic)
557:Hodge decomposition
48:Hodge decomposition
1534:Spectral sequences
1382:10.1007/bf01389078
1267:10.1007/BF02698925
1220:10.1007/BF02698925
1121:
894:
863:
828:
798:
694:
655:
589:
545:
526:projective variety
499:
469:
367:
247:differential forms
232:
205:
144:
1529:Complex manifolds
1012:perfect complexes
568:
336:
310:
64:spectral sequence
1541:
1510:
1485:
1476:(6): 2675–2708,
1464:
1455:
1438:
1413:
1392:
1345:
1336:
1310:
1278:
1277:
1255:Publ. Math. IHÉS
1246:
1240:
1239:
1199:
1193:
1187:
1130:
1128:
1127:
1122:
1111:
1100:
1099:
1081:
1080:
1059:
1048:
1047:
903:
901:
900:
895:
893:
892:
872:
870:
869:
864:
862:
861:
837:
835:
834:
829:
827:
826:
807:
805:
804:
799:
791:
790:
772:
771:
750:
749:
731:
730:
703:
701:
700:
695:
664:
662:
661:
656:
648:
634:
633:
618:
617:
599:
598:
588:
554:
552:
551:
546:
544:
543:
524:, for example a
508:
506:
505:
500:
498:
497:
478:
476:
475:
470:
459:
458:
440:
439:
412:
411:
402:
401:
382:Hodge filtration
376:
374:
373:
368:
360:
359:
338:
337:
335:
330:
325:
322:
321:
312:
311:
309:
304:
299:
296:
295:
280:
279:
267:
241:
239:
238:
233:
231:
230:
214:
212:
211:
206:
201:
187:
186:
163:complex manifold
153:
151:
150:
145:
140:
126:
125:
104:
103:
85:
84:
44:complex manifold
36:Alfred Frölicher
1549:
1548:
1544:
1543:
1542:
1540:
1539:
1538:
1514:
1513:
1467:
1441:
1395:
1350:Deligne, Pierre
1348:
1290:
1287:
1282:
1281:
1261:(35): 259–278,
1248:
1247:
1243:
1201:
1200:
1196:
1188:
1184:
1179:
1156:
1091:
1069:
1039:
1031:
1030:
1006:, and also its
996:
969:
952:
884:
879:
878:
875:de Rham complex
853:
848:
847:
818:
813:
812:
782:
757:
741:
722:
717:
716:
710:
674:
673:
625:
609:
590:
563:
562:
535:
530:
529:
519:Kähler manifold
515:
489:
484:
483:
444:
431:
403:
393:
388:
387:
345:
313:
287:
271:
258:
257:
222:
217:
216:
172:
167:
166:
111:
95:
76:
71:
70:
66:is as follows:
60:
26:Georges de Rham
12:
11:
5:
1547:
1545:
1537:
1536:
1531:
1526:
1516:
1515:
1512:
1511:
1465:
1439:
1404:(3): 785–876,
1393:
1368:(2): 247–270,
1346:
1301:(9): 641–644,
1286:
1283:
1280:
1279:
1241:
1214:(1): 107–126.
1194:
1181:
1180:
1178:
1175:
1174:
1173:
1170:Jacobian ideal
1167:
1162:
1155:
1152:
1140:Kaledin (2016)
1136:Kaledin (2008)
1132:
1131:
1120:
1117:
1114:
1110:
1106:
1103:
1098:
1094:
1090:
1087:
1084:
1079:
1076:
1072:
1068:
1065:
1062:
1058:
1054:
1051:
1046:
1042:
1038:
995:
992:
965:
950:
891:
887:
860:
856:
825:
821:
809:
808:
797:
794:
789:
785:
781:
778:
775:
770:
767:
764:
760:
756:
753:
748:
744:
740:
737:
734:
729:
725:
709:
706:
693:
690:
687:
684:
681:
666:
665:
654:
651:
647:
643:
640:
637:
632:
628:
624:
621:
616:
612:
608:
605:
602:
597:
593:
587:
584:
581:
578:
575:
571:
542:
538:
514:
511:
496:
492:
480:
479:
468:
465:
462:
457:
454:
451:
447:
443:
438:
434:
430:
427:
424:
421:
418:
415:
410:
406:
400:
396:
378:
377:
366:
363:
358:
355:
352:
348:
344:
341:
334:
329:
320:
316:
308:
303:
294:
290:
286:
283:
278:
274:
270:
266:
251:Poincaré lemma
229:
225:
204:
200:
196:
193:
190:
185:
182:
179:
175:
155:
154:
143:
139:
135:
132:
129:
124:
121:
118:
114:
110:
107:
102:
98:
94:
91:
88:
83:
79:
59:
56:
22:W. V. D. Hodge
13:
10:
9:
6:
4:
3:
2:
1546:
1535:
1532:
1530:
1527:
1525:
1522:
1521:
1519:
1509:
1505:
1501:
1497:
1493:
1489:
1484:
1479:
1475:
1471:
1466:
1463:
1459:
1454:
1449:
1445:
1440:
1437:
1433:
1429:
1425:
1421:
1417:
1412:
1407:
1403:
1399:
1394:
1391:
1387:
1383:
1379:
1375:
1371:
1367:
1363:
1362:Invent. Math.
1359:
1355:
1351:
1347:
1344:
1340:
1335:
1330:
1326:
1322:
1318:
1314:
1309:
1304:
1300:
1296:
1295:
1289:
1288:
1284:
1276:
1272:
1268:
1264:
1260:
1256:
1252:
1245:
1242:
1237:
1233:
1229:
1225:
1221:
1217:
1213:
1210:(in French).
1209:
1205:
1198:
1195:
1192:
1186:
1183:
1176:
1171:
1168:
1166:
1163:
1161:
1158:
1157:
1153:
1151:
1149:
1145:
1144:Mathew (2020)
1141:
1137:
1118:
1112:
1108:
1104:
1096:
1092:
1088:
1077:
1074:
1070:
1060:
1056:
1052:
1044:
1040:
1036:
1029:
1028:
1027:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
1000:dg categories
993:
991:
989:
985:
981:
977:
973:
968:
964:
960:
956:
948:
944:
940:
936:
932:
928:
925:
924:perfect field
921:
917:
913:
911:
907:
889:
876:
858:
845:
841:
823:
795:
787:
779:
776:
768:
765:
762:
758:
746:
738:
735:
727:
723:
715:
714:
713:
707:
705:
691:
685:
682:
679:
671:
652:
641:
638:
630:
626:
622:
614:
606:
603:
595:
591:
585:
582:
579:
576:
573:
569:
561:
560:
559:
558:
540:
536:
527:
523:
520:
512:
510:
494:
463:
455:
452:
449:
436:
425:
419:
413:
408:
398:
394:
386:
385:
384:
383:
364:
356:
353:
350:
339:
332:
318:
306:
292:
281:
276:
256:
255:
254:
252:
248:
245:
227:
223:
194:
191:
183:
180:
177:
173:
164:
160:
133:
130:
122:
119:
116:
112:
100:
92:
89:
81:
77:
69:
68:
67:
65:
57:
55:
53:
49:
45:
41:
37:
34:(named after
33:
32:
27:
23:
19:
1473:
1469:
1443:
1411:math/0611623
1401:
1397:
1365:
1361:
1357:
1354:Illusie, Luc
1298:
1292:
1258:
1254:
1244:
1211:
1207:
1197:
1190:
1185:
1165:Hodge theory
1133:
1023:
1019:
1015:
997:
987:
983:
975:
971:
966:
962:
958:
954:
947:Witt vectors
942:
938:
934:
930:
926:
919:
914:
909:
843:
810:
711:
670:Hodge theory
667:
521:
516:
513:Degeneration
481:
381:
379:
158:
156:
61:
54:cohomology.
29:
17:
15:
244:holomorphic
1518:Categories
1483:1710.09045
1453:1601.00637
1285:References
1508:119591893
1390:119635574
1275:121086388
1236:121086388
1228:0073-8301
1097:∗
1086:⇒
1075:±
1045:∗
886:Ω
859:∗
855:Ω
820:Ω
788:∗
784:Ω
755:⇒
743:Ω
689:→
611:Ω
570:⨁
495:∗
491:Ω
464:⋯
461:→
446:Ω
442:→
433:Ω
429:→
423:→
420:⋯
409:∗
405:Ω
354:
347:Ω
343:→
340:⋯
328:→
315:Ω
302:→
289:Ω
277:∗
273:Ω
269:→
109:⇒
97:Ω
1436:16703870
1343:16589720
1154:See also
1488:Bibcode
1458:Bibcode
1428:2435845
1370:Bibcode
1325:0073262
1177:Sources
922:over a
912:>0.
52:de Rham
50:of the
1506:
1434:
1426:
1388:
1341:
1334:528153
1331:
1323:
1315:
1273:
1234:
1226:
811:where
157:where
1504:S2CID
1478:arXiv
1448:arXiv
1432:S2CID
1406:arXiv
1386:S2CID
1317:89147
1313:JSTOR
1271:S2CID
1232:S2CID
937:)<
842:) on
161:is a
1339:PMID
1224:ISSN
1138:and
941:and
62:The
24:and
1496:doi
1416:doi
1378:doi
1329:PMC
1303:doi
1263:doi
1216:doi
482:of
351:dim
1520::
1502:,
1494:,
1486:,
1474:24
1472:,
1456:,
1446:,
1430:,
1424:MR
1422:,
1414:,
1400:,
1384:,
1376:,
1366:89
1364:,
1352:;
1337:,
1327:,
1321:MR
1319:,
1311:,
1299:41
1297:,
1269:,
1259:35
1257:,
1253:,
1230:.
1222:.
1212:35
1206:.
1150:.
846:,
509:.
414::=
282::=
165:,
1498::
1490::
1480::
1460::
1450::
1418::
1408::
1402:4
1380::
1372::
1358:p
1305::
1265::
1238:.
1218::
1119:.
1116:)
1113:k
1109:/
1105:C
1102:(
1093:P
1089:H
1083:]
1078:1
1071:u
1067:[
1064:)
1061:k
1057:/
1053:C
1050:(
1041:H
1037:H
1024:C
1020:X
1016:X
988:X
984:p
976:p
974:/
972:Z
967:p
963:F
961:=
959:k
955:k
953:(
951:2
949:W
943:X
939:p
935:X
931:p
927:k
920:X
910:p
890:q
844:X
824:p
796:,
793:)
780:,
777:X
774:(
769:q
766:+
763:p
759:H
752:)
747:p
739:,
736:X
733:(
728:q
724:H
692:S
686:X
683::
680:f
653:.
650:)
646:C
642:,
639:X
636:(
631:n
627:H
623:=
620:)
615:q
607:,
604:X
601:(
596:p
592:H
586:n
583:=
580:q
577:+
574:p
541:1
537:E
522:X
467:]
456:1
453:+
450:p
437:p
426:0
417:[
399:p
395:F
365:,
362:]
357:X
333:d
319:1
307:d
293:0
285:[
265:C
228:1
224:E
203:)
199:C
195:,
192:X
189:(
184:q
181:+
178:p
174:H
159:X
142:)
138:C
134:,
131:X
128:(
123:q
120:+
117:p
113:H
106:)
101:p
93:,
90:X
87:(
82:q
78:H
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.