130:). Convergent isocrystals are a variation of isocrystals that work better over non-perfect fields, and overconvergent isocrystals are another variation related to overconvergent cohomology theories.
653:
486:
436:
192:
112:
726:
331:
692:
772:
614:
541:
287:
397:
903:
883:
863:
843:
823:
803:
589:
569:
506:
371:
351:
252:
232:
212:
81:
1187:
1033:
1265:
994:
960:
1116:
1137:
936:
43:), who named them crystals because in some sense they are "rigid" and "grow". In particular quasicoherent crystals over the
1238:(2009), "p-adic cohomology", in Abramovich, Dan; Bertram, A.; Katzarkov, L.; Pandharipande, Rahul; Thaddeus., M. (eds.),
1291:
986:
1038:
1071:
36:
140:
619:
452:
402:
158:
88:
1253:
135:
52:
48:
1205:
733:
115:
1063:
915:
Ogus, Arthur (1 December 1984). "F-isocrystals and de Rham cohomology II—Convergent isocrystals".
1243:
1147:, Advanced studies in pure mathematics, vol. 3, Amsterdam: North-Holland, pp. 306–358,
442:
in the sense that they can be extended from open sets to infinitesimal extensions of open sets.
32:
1242:, Proc. Sympos. Pure Math., vol. 80, Providence, R.I.: Amer. Math. Soc., pp. 667–684,
1174:, Proc. Sympos. Pure Math., vol. 29, Providence, R.I.: Amer. Math. Soc., pp. 459–478,
292:
658:
1261:
1095:
1047:
990:
956:
28:
905:, then crystals of this fibered category are the same as crystals of the infinitesimal site.
742:
511:
257:
1223:, Proc. Sympos. Pure Math., vol. 55, Providence, RI: Amer. Math. Soc., pp. 43–70,
1087:
1020:
948:
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376:
44:
1275:
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1201:
1179:
1152:
1107:
1059:
1004:
970:
697:
1271:
1224:
1197:
1193:
1175:
1148:
1103:
1055:
1000:
966:
944:
825:, then a crystal is a cartesian section of the fibered category. In the special case when
1257:
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888:
868:
848:
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808:
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574:
554:
491:
356:
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237:
217:
197:
66:
1156:
1075:
1285:
1235:
63:
1143:, in Giraud, Jean; Grothendieck, Alexander; Kleiman, Steven L.; et al. (eds.),
17:
928:
1167:
1011:
Berthelot, P.; Ogus, A. (June 1983). "F-isocrystals and de Rham cohomology. I".
978:
126:(though the definition of isocrystal only appears in part II of this paper by
1099:
1051:
144:
943:, Lecture Notes in Mathematics, Vol. 407, vol. 407, Berlin, New York:
1186:
Illusie, Luc (1976), "Cohomologie cristalline (d'après P. Berthelot)",
1091:
1024:
952:
1248:
1080:
Institut des Hautes Études
Scientifiques. Publications Mathématiques
147:
is a structure in semilinear algebra somewhat related to crystals.
16:
For crystal graphs in representation theory of quantum groups, see
1189:
SĂ©minaire
Bourbaki (1974/1975: Exposés Nos. 453-470), Exp. No. 456
777:
Crystals on the crystalline site are defined in a similar way.
941:
Cohomologie cristalline des schémas de caractéristique p>0
194:
has as objects the infinitesimal extensions of open sets of
1192:, Lecture Notes in Math., vol. 514, Berlin, New York:
845:
is the category of infinitesimal extensions of a scheme
885:
the category of quasicoherent modules over objects of
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871:
851:
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811:
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161:
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Crystals over the infinitesimal and crystalline sites
91:
69:
1076:"On the de Rham cohomology of algebraic varieties"
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365:
345:
325:
281:
246:
226:
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186:
106:
75:
1138:"Crystals and the de Rham cohomology of schemes"
1219:Illusie, Luc (1994), "Crystalline cohomology",
123:
119:
40:
8:
1170:(1975), "Report on crystalline cohomology",
1145:Dix Exposés sur la Cohomologie des Schémas
1247:
1240:Algebraic geometry---Seattle 2005. Part 2
890:
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98:
93:
90:
68:
732:This is similar to the definition of a
739:An example of a crystal is the sheaf
62:is a crystal up to isogeny. They are
7:
1034:"Cohomologie cristalline: un survol"
736:of modules in the Zariski topology.
127:
373:as an abbreviation for an object
14:
1115:Grothendieck, Alexander (1966b),
648:{\displaystyle {\text{Inf}}(X/S)}
481:{\displaystyle {\text{Inf}}(X/S)}
431:{\displaystyle {\text{Inf}}(X/S)}
187:{\displaystyle {\text{Inf}}(X/S)}
1136:Grothendieck, Alexander (1968),
1111:(letter to Atiyah, Oct. 14 1963)
107:{\displaystyle \mathbf {Q} _{l}}
94:
1032:Chambert-Loir, Antoine (1998),
983:Notes on crystalline cohomology
47:are analogous to quasicoherent
781:Crystals in fibered categories
715:
704:
681:
675:
642:
628:
475:
461:
425:
411:
383:
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181:
167:
1:
929:10.1215/S0012-7094-84-05136-6
1221:Motives (Seattle, WA, 1991)
805:is a fibered category over
124:Berthelot & Ogus (1983)
35:. They were introduced by
1308:
987:Princeton University Press
326:{\displaystyle O_{X/S}(T)}
37:Alexander Grothendieck
15:
1039:Expositiones Mathematicae
917:Duke Mathematical Journal
687:{\displaystyle f^{*}F(T)}
1013:Inventiones Mathematicae
547:in the following sense:
1072:Grothendieck, Alexander
767:{\displaystyle O_{X/S}}
655:, the natural map from
536:{\displaystyle O_{X/S}}
438:. Sheaves on this site
282:{\displaystyle O_{X/S}}
155:The infinitesimal site
143:and Frobenius maps. An
899:
879:
859:
839:
819:
799:
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649:
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393:
392:{\displaystyle U\to T}
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108:
77:
900:
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769:
723:
721:{\displaystyle F(T')}
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333:= coordinate ring of
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120:Grothendieck (1966a)
89:
67:
1258:2006math......1507K
977:Berthelot, Pierre;
734:quasicoherent sheaf
1292:Algebraic geometry
1196:, pp. 53–60,
1172:Algebraic geometry
1092:10.1007/BF02684807
1025:10.1007/BF01389319
953:10.1007/BFb0068636
895:
875:
855:
835:
815:
795:
764:
728:is an isomorphism.
718:
684:
645:
609:{\displaystyle T'}
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389:
363:
343:
323:
279:
244:
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139:is a crystal with
104:
73:
33:fibered categories
29:Cartesian sections
1267:978-0-8218-4703-9
1236:Kedlaya, Kiran S.
1118:Letter to J. Tate
996:978-0-691-08218-9
962:978-3-540-06852-5
937:Berthelot, Pierre
898:{\displaystyle F}
878:{\displaystyle E}
858:{\displaystyle X}
838:{\displaystyle F}
818:{\displaystyle F}
798:{\displaystyle E}
626:
584:{\displaystyle T}
564:{\displaystyle f}
501:{\displaystyle F}
459:
409:
366:{\displaystyle T}
353:, where we write
346:{\displaystyle T}
247:{\displaystyle S}
234:is a scheme over
227:{\displaystyle X}
207:{\displaystyle X}
165:
136:Dieudonné crystal
118:, introduced by
76:{\displaystyle p}
1299:
1278:
1251:
1231:
1215:
1214:
1213:
1204:, archived from
1182:
1163:
1161:
1155:, archived from
1142:
1132:
1130:
1124:, archived from
1123:
1110:
1067:
1062:, archived from
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571:between objects
570:
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543:modules that is
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45:crystalline site
23:In mathematics,
1307:
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1301:
1300:
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1268:
1234:
1218:
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1194:Springer-Verlag
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1128:
1121:
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997:
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945:Springer-Verlag
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911:
887:
886:
867:
866:
847:
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786:
785:In general, if
783:
746:
741:
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400:
375:
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355:
354:
335:
334:
296:
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290:
289:is defined by
261:
256:
255:
254:then the sheaf
236:
235:
216:
215:
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12:
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1112:
1086:(29): 95–103,
1068:
1046:(4): 333–382,
1029:
1019:(2): 159–199.
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1241:
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1208:on 2012-02-10
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1199:
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1184:
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1173:
1169:
1165:
1162:on 2022-02-08
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1139:
1134:
1131:on 2021-07-21
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1066:on 2011-07-21
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85:analogues of
84:
70:
61:
56:
54:
50:
46:
42:
38:
34:
30:
26:
19:
1249:math/0601507
1239:
1220:
1210:, retrieved
1206:the original
1188:
1171:
1168:Illusie, Luc
1157:the original
1144:
1126:the original
1117:
1083:
1079:
1064:the original
1043:
1037:
1016:
1012:
982:
979:Ogus, Arthur
940:
920:
916:
784:
776:
738:
731:
551:for any map
544:
449:on the site
446:
444:
439:
154:
141:Verschiebung
134:
132:
114:-adic Ă©tale
59:
57:
24:
22:
18:crystal base
488:is a sheaf
128:Ogus (1984)
31:of certain
1212:2016-08-24
909:References
60:isocrystal
1100:0073-8301
1074:(1966a),
1052:0723-0869
668:∗
384:→
145:F-crystal
1286:Category
981:(1978),
939:(1974),
712:′
603:′
25:crystals
1276:2483951
1254:Bibcode
1229:1265522
1202:0444668
1180:0393034
1153:0269663
1108:0199194
1060:1654786
1005:0491705
971:0384804
447:crystal
116:sheaves
51:over a
49:modules
39: (
1274:
1264:
1227:
1200:
1178:
1151:
1106:
1098:
1058:
1050:
1003:
993:
969:
959:
53:scheme
1244:arXiv
1160:(PDF)
1141:(PDF)
1129:(PDF)
1122:(PDF)
923:(4).
616:; of
545:rigid
214:. If
83:-adic
41:1966a
1262:ISBN
1096:ISSN
1048:ISSN
991:ISBN
957:ISBN
865:and
440:grow
122:and
27:are
1088:doi
1021:doi
949:doi
925:doi
694:to
625:Inf
508:of
458:Inf
408:Inf
399:of
164:Inf
58:An
55:.
1288::
1272:MR
1270:,
1260:,
1252:,
1225:MR
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1094:,
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1078:,
1056:MR
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1044:16
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1036:,
1017:72
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999:,
989:,
985:,
967:MR
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955:,
947:,
921:51
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774:.
591:,
445:A
133:A
1256::
1246::
1090::
1027:.
1023::
951::
931:.
927::
893:F
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833:F
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756:/
752:X
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709:T
705:(
702:F
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679:T
676:(
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640:S
636:/
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462:(
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315:(
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263:O
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179:S
175:/
171:X
168:(
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71:p
20:.
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