664:
413:
659:{\displaystyle {\begin{matrix}&&\vdots &&\vdots &&\\&&\uparrow &&\uparrow &&\\\cdots &\to &C_{p,q+1}&\to &C_{p+1,q+1}&\to &\cdots \\&&\uparrow &&\uparrow &&\\\cdots &\to &C_{p,q}&\to &C_{p+1,q}&\to &\cdots \\&&\uparrow &&\uparrow &&\\&&\vdots &&\vdots &&\\\end{matrix}}}
872:
1187:
1083:
173:
745:
402:
92:
874:, we can switch between having commutativity and anticommutativity. If the commutative definition is used, this alternating sign will have to show up in the definition of Total Complexes.
330:
250:
1029:
960:
119:
62:
1000:
1115:
918:
1241:
757:
1123:
1301:
1034:
124:
678:
1321:
1316:
341:
67:
1289:
264:
184:
898:
94:-grading. The most general definition of a double complex, or a bicomplex, is given with objects in an
1005:
923:
751:
100:
31:
45:
1266:
965:
887:
1247:
1237:
95:
882:
There are many natural examples of bicomplexes that come up in nature. In particular, for a
20:
1265:
Block, Jonathan; Daenzer, Calder (2009-01-09). "Mukai duality for gerbes with connection".
1088:
1302:
https://web.archive.org/web/20210708183754/http://www.dma.unifi.it/~vezzosi/papers/tou.pdf
1207:"Section 12.18 (0FNB): Double complexes and associated total complexes—The Stacks project"
903:
1310:
1284:
39:
894:
883:
27:
1251:
1206:
886:, there is a bicomplex associated to it which can be used to construct its
1231:
867:{\displaystyle f_{p,q}=(-1)^{p}d_{p,q}^{v}\colon C_{p,q}\to C_{p,q-1}}
672:
Some authors instead require that the squares anticommute. That is
1271:
962:
whose components are linear or anti-linear. For example, if
157:
106:
407:
Hence a double complex is a commutative diagram of the form
1182:{\displaystyle f_{a,b}dz_{a}\wedge d{\overline {z}}_{b}}
19:"Bicomplex" redirects here. For the type of number, see
1078:{\displaystyle {\overline {z}}_{1},{\overline {z}}_{2}}
418:
168:{\displaystyle C_{p,q}\in {\text{Ob}}({\mathcal {A}})}
1126:
1091:
1037:
1008:
968:
926:
906:
760:
681:
416:
344:
267:
187:
127:
103:
70:
48:
740:{\displaystyle d_{h}\circ d_{v}+d_{v}\circ d_{h}=0.}
1181:
1109:
1085:are the complex conjugate of these coordinates, a
1077:
1023:
994:
954:
912:
866:
739:
658:
396:
324:
244:
167:
113:
86:
56:
669:where the rows and columns form chain complexes.
397:{\displaystyle d_{h}\circ d_{v}=d_{v}\circ d_{h}}
87:{\displaystyle \mathbb {Z} \times \mathbb {Z} }
64:-grading, the objects in the bicomplex have a
893:Another common example of bicomplexes are in
8:
1236:. Cambridge : Cambridge University Press.
920:there's a bicomplex of differential forms
325:{\displaystyle d^{v}:C_{p,q}\to C_{p,q+1}}
245:{\displaystyle d^{h}:C_{p,q}\to C_{p+1,q}}
1270:
1173:
1163:
1150:
1131:
1125:
1090:
1069:
1059:
1049:
1039:
1036:
1015:
1011:
1010:
1007:
986:
973:
967:
931:
925:
905:
846:
827:
814:
803:
793:
765:
759:
725:
712:
699:
686:
680:
590:
567:
498:
469:
417:
415:
388:
375:
362:
349:
343:
304:
285:
272:
266:
224:
205:
192:
186:
156:
155:
147:
132:
126:
105:
104:
102:
80:
79:
72:
71:
69:
50:
49:
47:
1198:
121:. A bicomplex is a sequence of objects
1233:An introduction to homological algebra
335:which have the compatibility relation
7:
928:
14:
1024:{\displaystyle \mathbb {C} ^{2}}
955:{\displaystyle \Omega ^{p,q}(X)}
1002:are the complex coordinates of
1104:
1092:
949:
943:
839:
790:
780:
630:
624:
610:
581:
558:
544:
538:
524:
489:
460:
446:
440:
297:
217:
162:
152:
114:{\displaystyle {\mathcal {A}}}
1:
750:This eases the definition of
1168:
1064:
1044:
175:with two differentials, the
57:{\displaystyle \mathbb {Z} }
1230:Weibel, Charles A. (1994).
995:{\displaystyle z_{1},z_{2}}
1338:
1290:Derived algebraic geometry
42:where instead of having a
18:
38:is a generalization of a
1211:stacks.math.columbia.edu
1296:Additional applications
899:almost complex manifold
177:horizontal differential
1190:
1183:
1111:
1079:
1025:
996:
956:
914:
868:
748:
741:
667:
660:
405:
398:
333:
326:
253:
246:
169:
115:
88:
58:
1184:
1119:
1112:
1110:{\displaystyle (1,1)}
1080:
1026:
997:
957:
915:
869:
742:
674:
661:
409:
399:
337:
327:
260:
257:vertical differential
247:
180:
170:
116:
89:
59:
1124:
1117:-form is of the form
1089:
1035:
1006:
966:
924:
904:
758:
679:
414:
342:
265:
185:
125:
101:
68:
46:
16:Mathematical concept
1322:Additive categories
1317:Homological algebra
819:
32:Homological algebra
1179:
1107:
1075:
1021:
992:
952:
910:
864:
799:
737:
656:
654:
394:
322:
242:
165:
111:
84:
54:
1243:978-1-139-64863-9
1171:
1067:
1047:
913:{\displaystyle X}
150:
96:additive category
1329:
1277:
1276:
1274:
1262:
1256:
1255:
1227:
1221:
1220:
1218:
1217:
1203:
1188:
1186:
1185:
1180:
1178:
1177:
1172:
1164:
1155:
1154:
1142:
1141:
1116:
1114:
1113:
1108:
1084:
1082:
1081:
1076:
1074:
1073:
1068:
1060:
1054:
1053:
1048:
1040:
1030:
1028:
1027:
1022:
1020:
1019:
1014:
1001:
999:
998:
993:
991:
990:
978:
977:
961:
959:
958:
953:
942:
941:
919:
917:
916:
911:
873:
871:
870:
865:
863:
862:
838:
837:
818:
813:
798:
797:
776:
775:
746:
744:
743:
738:
730:
729:
717:
716:
704:
703:
691:
690:
665:
663:
662:
657:
655:
652:
651:
645:
639:
638:
635:
634:
628:
622:
621:
607:
606:
578:
577:
549:
548:
542:
536:
535:
521:
520:
486:
485:
451:
450:
444:
438:
437:
434:
433:
427:
421:
420:
403:
401:
400:
395:
393:
392:
380:
379:
367:
366:
354:
353:
331:
329:
328:
323:
321:
320:
296:
295:
277:
276:
251:
249:
248:
243:
241:
240:
216:
215:
197:
196:
174:
172:
171:
166:
161:
160:
151:
148:
143:
142:
120:
118:
117:
112:
110:
109:
93:
91:
90:
85:
83:
75:
63:
61:
60:
55:
53:
21:Bicomplex number
1337:
1336:
1332:
1331:
1330:
1328:
1327:
1326:
1307:
1306:
1298:
1281:
1280:
1264:
1263:
1259:
1244:
1229:
1228:
1224:
1215:
1213:
1205:
1204:
1200:
1195:
1162:
1146:
1127:
1122:
1121:
1087:
1086:
1058:
1038:
1033:
1032:
1009:
1004:
1003:
982:
969:
964:
963:
927:
922:
921:
902:
901:
888:de-Rham complex
880:
842:
823:
789:
761:
756:
755:
752:Total Complexes
721:
708:
695:
682:
677:
676:
653:
650:
644:
636:
633:
627:
619:
618:
613:
608:
586:
584:
579:
563:
561:
556:
550:
547:
541:
533:
532:
527:
522:
494:
492:
487:
465:
463:
458:
452:
449:
443:
435:
432:
426:
412:
411:
384:
371:
358:
345:
340:
339:
300:
281:
268:
263:
262:
220:
201:
188:
183:
182:
128:
123:
122:
99:
98:
66:
65:
44:
43:
30:, specifically
24:
17:
12:
11:
5:
1335:
1333:
1325:
1324:
1319:
1309:
1308:
1305:
1304:
1297:
1294:
1293:
1292:
1287:
1279:
1278:
1257:
1242:
1222:
1197:
1196:
1194:
1191:
1176:
1170:
1167:
1161:
1158:
1153:
1149:
1145:
1140:
1137:
1134:
1130:
1106:
1103:
1100:
1097:
1094:
1072:
1066:
1063:
1057:
1052:
1046:
1043:
1018:
1013:
989:
985:
981:
976:
972:
951:
948:
945:
940:
937:
934:
930:
909:
897:, where on an
879:
876:
861:
858:
855:
852:
849:
845:
841:
836:
833:
830:
826:
822:
817:
812:
809:
806:
802:
796:
792:
788:
785:
782:
779:
774:
771:
768:
764:
736:
733:
728:
724:
720:
715:
711:
707:
702:
698:
694:
689:
685:
649:
646:
643:
640:
637:
632:
629:
626:
623:
620:
617:
614:
612:
609:
605:
602:
599:
596:
593:
589:
585:
583:
580:
576:
573:
570:
566:
562:
560:
557:
555:
552:
551:
546:
543:
540:
537:
534:
531:
528:
526:
523:
519:
516:
513:
510:
507:
504:
501:
497:
493:
491:
488:
484:
481:
478:
475:
472:
468:
464:
462:
459:
457:
454:
453:
448:
445:
442:
439:
436:
431:
428:
425:
422:
419:
391:
387:
383:
378:
374:
370:
365:
361:
357:
352:
348:
319:
316:
313:
310:
307:
303:
299:
294:
291:
288:
284:
280:
275:
271:
239:
236:
233:
230:
227:
223:
219:
214:
211:
208:
204:
200:
195:
191:
164:
159:
154:
146:
141:
138:
135:
131:
108:
82:
78:
74:
52:
36:double complex
15:
13:
10:
9:
6:
4:
3:
2:
1334:
1323:
1320:
1318:
1315:
1314:
1312:
1303:
1300:
1299:
1295:
1291:
1288:
1286:
1285:Chain complex
1283:
1282:
1273:
1268:
1261:
1258:
1253:
1249:
1245:
1239:
1235:
1234:
1226:
1223:
1212:
1208:
1202:
1199:
1192:
1189:
1174:
1165:
1159:
1156:
1151:
1147:
1143:
1138:
1135:
1132:
1128:
1118:
1101:
1098:
1095:
1070:
1061:
1055:
1050:
1041:
1016:
987:
983:
979:
974:
970:
946:
938:
935:
932:
907:
900:
896:
891:
889:
885:
877:
875:
859:
856:
853:
850:
847:
843:
834:
831:
828:
824:
820:
815:
810:
807:
804:
800:
794:
786:
783:
777:
772:
769:
766:
762:
754:. By setting
753:
747:
734:
731:
726:
722:
718:
713:
709:
705:
700:
696:
692:
687:
683:
673:
670:
666:
647:
641:
615:
603:
600:
597:
594:
591:
587:
574:
571:
568:
564:
553:
529:
517:
514:
511:
508:
505:
502:
499:
495:
482:
479:
476:
473:
470:
466:
455:
429:
423:
408:
404:
389:
385:
381:
376:
372:
368:
363:
359:
355:
350:
346:
336:
332:
317:
314:
311:
308:
305:
301:
292:
289:
286:
282:
278:
273:
269:
259:
258:
252:
237:
234:
231:
228:
225:
221:
212:
209:
206:
202:
198:
193:
189:
179:
178:
144:
139:
136:
133:
129:
97:
76:
41:
40:chain complex
37:
33:
29:
22:
1260:
1232:
1225:
1214:. Retrieved
1210:
1201:
1120:
895:Hodge theory
892:
884:Lie groupoid
881:
749:
675:
671:
668:
410:
406:
338:
334:
261:
256:
254:
181:
176:
35:
25:
28:mathematics
1311:Categories
1216:2021-07-08
1272:0803.1529
1252:847527211
1169:¯
1157:∧
1065:¯
1045:¯
929:Ω
857:−
840:→
821::
784:−
719:∘
693:∘
648:⋮
642:⋮
631:↑
625:↑
616:⋯
611:→
582:→
559:→
554:⋯
545:↑
539:↑
530:⋯
525:→
490:→
461:→
456:⋯
447:↑
441:↑
430:⋮
424:⋮
382:∘
356:∘
298:→
218:→
145:∈
77:×
1193:See also
878:Examples
255:and the
1250:
1240:
1267:arXiv
1248:OCLC
1238:ISBN
1031:and
34:, a
26:In
1313::
1246:.
1209:.
890:.
735:0.
149:Ob
1275:.
1269::
1254:.
1219:.
1175:b
1166:z
1160:d
1152:a
1148:z
1144:d
1139:b
1136:,
1133:a
1129:f
1105:)
1102:1
1099:,
1096:1
1093:(
1071:2
1062:z
1056:,
1051:1
1042:z
1017:2
1012:C
988:2
984:z
980:,
975:1
971:z
950:)
947:X
944:(
939:q
936:,
933:p
908:X
860:1
854:q
851:,
848:p
844:C
835:q
832:,
829:p
825:C
816:v
811:q
808:,
805:p
801:d
795:p
791:)
787:1
781:(
778:=
773:q
770:,
767:p
763:f
732:=
727:h
723:d
714:v
710:d
706:+
701:v
697:d
688:h
684:d
604:q
601:,
598:1
595:+
592:p
588:C
575:q
572:,
569:p
565:C
518:1
515:+
512:q
509:,
506:1
503:+
500:p
496:C
483:1
480:+
477:q
474:,
471:p
467:C
390:h
386:d
377:v
373:d
369:=
364:v
360:d
351:h
347:d
318:1
315:+
312:q
309:,
306:p
302:C
293:q
290:,
287:p
283:C
279::
274:v
270:d
238:q
235:,
232:1
229:+
226:p
222:C
213:q
210:,
207:p
203:C
199::
194:h
190:d
163:)
158:A
153:(
140:q
137:,
134:p
130:C
107:A
81:Z
73:Z
51:Z
23:.
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