929:
396:
244:
683:. In a different direction, one can dispense with such assumptions entirely if one instead works with derived de Rham cohomology (now taking the associated graded of the conjugate filtration) and the exterior powers of the
846:
186:
656:
The smoothness assumption is not essential for the
Cartier map to be an isomorphism. For instance, one has it for ind-smooth morphisms since both sides of the Cartier map commute with
525:
440:
560:
231:
970:
798:
63:. Intuitively, it shows that de Rham cohomology in positive characteristic is a much larger object than one might expect. It plays an important role in the approach of
651:
594:
121:
614:
963:
391:{\displaystyle C^{-1}:\bigoplus _{i\geq 0}\Omega _{X^{(p)}/S}^{i}\to \bigoplus _{i\geq 0}{\mathcal {H}}^{i}(F_{*}\Omega _{X/S}^{\bullet })}
72:
912:
994:
562:. (Here, for the Cartier map to be well-defined in general it is essential that one takes cohomology sheaves for the codomain.) The
956:
619:
In the above, we have formulated the
Cartier isomorphism in the form it is most commonly encountered (e.g., in the 1970 paper of
60:
623:). In his original paper, Cartier actually considered the inverse map in a more restrictive setting, whence the notation
33:
989:
133:
445:
48:
401:
234:
56:
534:
29:
661:
124:
704:
Pierre
Deligne; Luc Illusie (1987). "Relèvements modulo p et décomposition du complexe de de Rham".
191:
44:
936:
884:
858:
826:
760:
721:
17:
876:
818:
684:
669:
52:
40:
868:
810:
752:
713:
665:
657:
626:
569:
904:
94:
741:"Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin"
940:
680:
599:
64:
983:
888:
830:
740:
725:
37:
764:
778:
Cartier, Pierre (1957). "Une nouvelle opération sur les formes différentielles".
928:
676:
68:
25:
872:
814:
880:
822:
620:
745:
Publications Mathématiques de l'Institut des Hautes Études
Scientifiques
756:
717:
863:
541:
408:
338:
845:
Kerz, Moritz; Strunk, Florian; Tamme, Georg (2021-05-20).
944:
847:"Towards Vorst's conjecture in positive characteristic"
404:
97:
629:
602:
572:
537:
448:
247:
194:
136:
645:
608:
588:
554:
519:
434:
390:
225:
180:
115:
181:{\displaystyle X^{(p)}=X\times _{S,\varphi }S}
964:
664:, one then has the Cartier isomorphism for a
8:
797:Kelly, Shane; Morrow, Matthew (2021-05-20).
971:
957:
679:has also proven a Cartier isomorphism for
862:
634:
628:
601:
577:
571:
546:
540:
539:
536:
496:
453:
447:
418:
413:
407:
406:
403:
379:
370:
366:
356:
343:
337:
336:
323:
310:
301:
289:
284:
268:
252:
246:
211:
193:
163:
141:
135:
96:
696:
520:{\displaystyle C^{-1}(d(x\otimes 1))=}
435:{\textstyle {\mathcal {O}}_{X^{(p)}}}
7:
925:
923:
241:is defined to be the unique morphism
566:is then the assertion that the map
188:denote the Frobenius twist and let
555:{\displaystyle {\mathcal {O}}_{X}}
363:
281:
59:of the variety. It is named after
14:
739:Nicholas M. Katz (January 1970).
927:
915:from the original on 2022-09-22.
73:Hodge–de Rham spectral sequence
514:
489:
483:
480:
468:
462:
425:
419:
385:
349:
316:
296:
290:
226:{\displaystyle F:X\to X^{(p)}}
218:
212:
204:
148:
142:
107:
1:
799:"K-theory of valuation rings"
87:be a field of characteristic
943:. You can help Knowledge by
905:"Derived de Rham cohomology"
71:to the degeneration of the
1011:
922:
873:10.1112/S0010437X21007120
815:10.1112/S0010437X21007119
995:Algebraic geometry stubs
706:Inventiones Mathematicae
49:positive characteristic
939:–related article is a
851:Compositio Mathematica
803:Compositio Mathematica
780:C. R. Acad. Sci. Paris
647:
646:{\displaystyle C^{-1}}
616:is a smooth morphism.
610:
590:
589:{\displaystyle C^{-1}}
556:
527:for any local section
521:
436:
392:
227:
182:
117:
835:cf. discussion in §2.
653:for the Cartier map.
648:
611:
596:is an isomorphism if
591:
557:
522:
437:
393:
228:
183:
118:
116:{\textstyle f:X\to S}
51:, and the sheaves of
627:
600:
570:
535:
446:
442:-algebras such that
402:
245:
192:
134:
95:
564:Cartier isomorphism
384:
315:
22:Cartier isomorphism
990:Algebraic geometry
937:algebraic geometry
903:Kedlaya, Kiran S.
757:10.1007/BF02684688
718:10.1007/BF01389078
643:
606:
586:
552:
517:
432:
388:
362:
334:
280:
279:
235:relative Frobenius
223:
178:
113:
53:differential forms
30:cohomology sheaves
18:algebraic geometry
952:
951:
685:cotangent complex
662:Popescu's theorem
658:filtered colimits
609:{\displaystyle f}
319:
264:
123:be a morphism of
41:algebraic variety
1002:
973:
966:
959:
931:
924:
917:
916:
911:. Prop. 17.2.4.
900:
894:
892:
866:
857:(6): 1143–1171.
842:
836:
834:
809:(6): 1121–1142.
794:
788:
787:
775:
769:
768:
736:
730:
729:
701:
666:regular morphism
652:
650:
649:
644:
642:
641:
615:
613:
612:
607:
595:
593:
592:
587:
585:
584:
561:
559:
558:
553:
551:
550:
545:
544:
526:
524:
523:
518:
507:
506:
461:
460:
441:
439:
438:
433:
431:
430:
429:
428:
412:
411:
397:
395:
394:
389:
383:
378:
374:
361:
360:
348:
347:
342:
341:
333:
314:
309:
305:
300:
299:
278:
260:
259:
232:
230:
229:
224:
222:
221:
187:
185:
184:
179:
174:
173:
152:
151:
122:
120:
119:
114:
91:> 0, and let
1010:
1009:
1005:
1004:
1003:
1001:
1000:
999:
980:
979:
978:
977:
921:
920:
902:
901:
897:
893:cf. Appendix A.
844:
843:
839:
796:
795:
791:
777:
776:
772:
738:
737:
733:
703:
702:
698:
693:
681:valuation rings
630:
625:
624:
598:
597:
573:
568:
567:
538:
533:
532:
492:
449:
444:
443:
414:
405:
400:
399:
352:
335:
285:
248:
243:
242:
207:
190:
189:
159:
137:
132:
131:
93:
92:
81:
57:Frobenius twist
34:de Rham complex
12:
11:
5:
1008:
1006:
998:
997:
992:
982:
981:
976:
975:
968:
961:
953:
950:
949:
932:
919:
918:
895:
837:
789:
770:
731:
712:(2): 247–270.
695:
694:
692:
689:
640:
637:
633:
605:
583:
580:
576:
549:
543:
516:
513:
510:
505:
502:
499:
495:
491:
488:
485:
482:
479:
476:
473:
470:
467:
464:
459:
456:
452:
427:
424:
421:
417:
410:
387:
382:
377:
373:
369:
365:
359:
355:
351:
346:
340:
332:
329:
326:
322:
318:
313:
308:
304:
298:
295:
292:
288:
283:
277:
274:
271:
267:
263:
258:
255:
251:
220:
217:
214:
210:
206:
203:
200:
197:
177:
172:
169:
166:
162:
158:
155:
150:
147:
144:
140:
112:
109:
106:
103:
100:
80:
77:
61:Pierre Cartier
13:
10:
9:
6:
4:
3:
2:
1007:
996:
993:
991:
988:
987:
985:
974:
969:
967:
962:
960:
955:
954:
948:
946:
942:
938:
933:
930:
926:
914:
910:
909:kskedlaya.org
906:
899:
896:
890:
886:
882:
878:
874:
870:
865:
860:
856:
852:
848:
841:
838:
832:
828:
824:
820:
816:
812:
808:
804:
800:
793:
790:
785:
781:
774:
771:
766:
762:
758:
754:
750:
746:
742:
735:
732:
727:
723:
719:
715:
711:
707:
700:
697:
690:
688:
686:
682:
678:
674:
671:
667:
663:
659:
654:
638:
635:
631:
622:
617:
603:
581:
578:
574:
565:
547:
530:
511:
508:
503:
500:
497:
493:
486:
477:
474:
471:
465:
457:
454:
450:
422:
415:
380:
375:
371:
367:
357:
353:
344:
330:
327:
324:
320:
311:
306:
302:
293:
286:
275:
272:
269:
265:
261:
256:
253:
249:
240:
236:
215:
208:
201:
198:
195:
175:
170:
167:
164:
160:
156:
153:
145:
138:
129:
127:
110:
104:
101:
98:
90:
86:
78:
76:
74:
70:
66:
62:
58:
54:
50:
46:
42:
39:
35:
31:
27:
24:is a certain
23:
19:
945:expanding it
934:
908:
898:
854:
850:
840:
806:
802:
792:
783:
779:
773:
748:
744:
734:
709:
705:
699:
672:
655:
618:
563:
528:
238:
125:
88:
84:
82:
28:between the
21:
15:
751:: 175–232.
677:Ofer Gabber
239:Cartier map
26:isomorphism
984:Categories
864:1812.05342
786:: 426–428.
691:References
675:-schemes.
670:noetherian
398:of graded
889:119755507
881:0010-437X
831:119721861
823:0010-437X
726:119635574
636:−
579:−
501:−
475:⊗
455:−
381:∙
364:Ω
358:∗
328:≥
321:⨁
317:→
282:Ω
273:≥
266:⨁
254:−
205:→
171:φ
161:×
108:→
79:Statement
913:Archived
765:16261793
128:-schemes
233:be the
69:Illusie
65:Deligne
55:on the
43:over a
32:of the
887:
879:
829:
821:
763:
724:
237:. The
130:. Let
38:smooth
20:, the
935:This
885:S2CID
859:arXiv
827:S2CID
761:S2CID
722:S2CID
660:. By
45:field
36:of a
941:stub
877:ISSN
819:ISSN
621:Katz
83:Let
67:and
869:doi
855:157
811:doi
807:157
784:244
753:doi
714:doi
668:of
531:of
47:of
16:In
986::
907:.
883:.
875:.
867:.
853:.
849:.
825:.
817:.
805:.
801:.
782:.
759:.
749:39
747:.
743:.
720:.
710:89
708:.
687:.
75:.
972:e
965:t
958:v
947:.
891:.
871::
861::
833:.
813::
767:.
755::
728:.
716::
673:k
639:1
632:C
604:f
582:1
575:C
548:X
542:O
529:x
515:]
512:x
509:d
504:1
498:p
494:x
490:[
487:=
484:)
481:)
478:1
472:x
469:(
466:d
463:(
458:1
451:C
426:)
423:p
420:(
416:X
409:O
386:)
376:S
372:/
368:X
354:F
350:(
345:i
339:H
331:0
325:i
312:i
307:S
303:/
297:)
294:p
291:(
287:X
276:0
270:i
262::
257:1
250:C
219:)
216:p
213:(
209:X
202:X
199::
196:F
176:S
168:,
165:S
157:X
154:=
149:)
146:p
143:(
139:X
126:k
111:S
105:X
102::
99:f
89:p
85:k
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.