1442:
893:
1583:
585:
Of particular interest to the theory of constructible Ă©tale sheaves is the case in which one works with constructible Ă©tale sheaves of
Abelian groups. The remarkable result is that constructible Ă©tale sheaves of Abelian groups are precisely the Noetherian objects in the category of all torsion Ă©tale
1309:
769:
1453:
302:
459:
1637:
373:
774:
667:
1437:{\displaystyle \mathbf {R} ^{0}\pi _{*}({\underline {\mathbb {Q} }}_{X})\cong \mathbf {R} ^{2}\pi _{*}({\underline {\mathbb {Q} }}_{X})\cong {\underline {\mathbb {Q} }}_{\mathbb {C} }}
1081:
952:
1301:
1210:
1016:
984:
1666:
1040:
920:
580:
536:
509:
162:
115:
1235:
221:
1113:
687:
1267:
1692:
888:{\displaystyle {\begin{aligned}T_{0}={\begin{bmatrix}1&k\\0&1\end{bmatrix}},\quad &T_{1}={\begin{bmatrix}1&l\\0&1\end{bmatrix}}\end{aligned}}}
717:
1578:{\displaystyle \mathbf {R} ^{1}\pi _{*}({\underline {\mathbb {Q} }}_{X})\cong {\mathcal {L}}_{\mathbb {C} -\{0,1\}}\oplus {\underline {\mathbb {Q} }}_{\{0,1\}}}
1133:
761:
741:
556:
482:
393:
322:
241:
182:
135:
464:
This definition allows us to derive, from
Noetherian induction and the fact that an Ă©tale sheaf is constant if and only if its restriction from
1714:
610:
One nice set of examples of constructible sheaves come from the derived pushforward (with or without compact support) of a local system on
246:
75:
1755:
Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 3
1812:
1778:
398:
1591:
327:
613:
1846:
1762:
1045:
925:
1272:
1695:
1872:
1152:
989:
957:
1642:
93:
from the book by
Freitag and Kiehl referenced below. In what follows in this subsection, all sheaves
78:
in Ă©tale cohomology states that the higher direct images of a constructible sheaf are constructible.
1021:
901:
561:
514:
487:
143:
96:
1218:
187:
1086:
595:
1723:
1833:, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 13, Berlin: Springer-Verlag,
1745:
48:
1877:
672:
44:
1753:
1856:
1822:
1788:
1240:
8:
1671:
696:
25:
56:
1749:
1118:
746:
726:
541:
467:
378:
307:
226:
167:
120:
52:
1842:
1808:
1774:
33:
1115:
compute the cohomology of the local systems restricted to a neighborhood of them in
1834:
1766:
68:
64:
1269:
this family of curves degenerates into a nodal curve. If we denote this family by
1852:
1818:
1804:
1796:
1784:
304:
is a finite locally constant sheaf. In particular, this means for each subscheme
1144:
1838:
90:
1866:
1761:. Lecture Notes in Mathematics (in French). Vol. 305. Berlin; New York:
1741:
29:
599:
81:
17:
1770:
720:
690:
602:
on a family of topological spaces parameterized by a base space.
297:{\displaystyle {\mathcal {F}}|_{Y}=i_{Y}^{\ast }{\mathcal {F}}}
184:
can be written as a finite union of locally closed subschemes
324:
appearing in the finite covering, there is an Ă©tale covering
1083:
we get a constructible sheaf where the stalks at the points
1668:. This local monodromy around of this local system around
1740:
1138:
763:. For example, we can set the monodromy operators to be
60:
850:
795:
454:{\displaystyle (i_{Y})^{\ast }{\mathcal {F}}|_{U_{i}}}
82:
Definition of Ă©tale constructible sheaves on a scheme
1674:
1645:
1632:{\displaystyle {\mathcal {L}}_{\mathbb {C} -\{0,1\}}}
1594:
1456:
1312:
1275:
1243:
1221:
1155:
1121:
1089:
1048:
1024:
992:
960:
928:
904:
772:
749:
729:
699:
675:
616:
564:
544:
517:
490:
470:
401:
381:
368:{\displaystyle \lbrace U_{i}\to Y\mid i\in I\rbrace }
330:
310:
249:
229:
190:
170:
146:
123:
99:
59:
constructible sheaves are defined in a similar way (
558:. It then follows that a representable Ă©tale sheaf
375:such that for all Ă©tale subschemes in the cover of
1686:
1660:
1631:
1577:
1436:
1295:
1261:
1229:
1204:
1127:
1107:
1075:
1034:
1010:
978:
946:
914:
887:
755:
735:
711:
681:
662:{\displaystyle U=\mathbb {P} ^{1}-\{0,1,\infty \}}
661:
586:sheaves (cf. Proposition I.4.8 of Freitag-Kiehl).
574:
550:
530:
503:
476:
453:
387:
367:
316:
296:
235:
215:
176:
156:
129:
109:
1143:For example, consider the family of degenerating
594:Most examples of constructible sheaves come from
589:
1864:
598:sheaves or from the derived pushforward of a
461:is constant and represented by a finite set.
1829:Freitag, Eberhard; Kiehl, Reinhardt (1988),
1828:
1712:
1624:
1612:
1570:
1558:
1536:
1524:
656:
638:
605:
362:
331:
89:Here we use the definition of constructible
954:. Then, if we take the derived pushforward
67:of constructible sheaves, see a section in
137:are Ă©tale sheaves unless otherwise noted.
1648:
1605:
1548:
1517:
1486:
1428:
1417:
1392:
1342:
1289:
1223:
1063:
931:
625:
1831:Etale Cohomology and the Weil Conjecture
1076:{\displaystyle j:U\to \mathbb {P} ^{1}}
947:{\displaystyle \mathbb {Q} ^{\oplus 2}}
1865:
1296:{\displaystyle \pi :X\to \mathbb {C} }
61:Artin, Grothendieck & Verdier 1972
1795:
1588:where the stalks of the local system
1139:Weierstrass family of elliptic curves
898:where the stalks of our local system
43:is the union of a finite number of
13:
1803:, Universitext, Berlin, New York:
1713:Gunningham, Sam; Hughes, Richard,
1598:
1510:
1102:
1027:
907:
676:
653:
567:
427:
289:
252:
149:
102:
14:
1889:
1205:{\displaystyle y^{2}-x(x-1)(x-t)}
1011:{\displaystyle \mathbf {R} j_{!}}
979:{\displaystyle \mathbf {R} j_{*}}
1706:
1661:{\displaystyle \mathbb {Q} ^{2}}
1459:
1365:
1315:
994:
962:
47:on each of which the sheaf is a
1734:
829:
538:is the reduction of the scheme
1501:
1479:
1407:
1385:
1357:
1335:
1285:
1199:
1187:
1184:
1172:
1058:
1035:{\displaystyle {\mathcal {L}}}
915:{\displaystyle {\mathcal {L}}}
590:Examples in algebraic topology
575:{\displaystyle {\mathcal {F}}}
531:{\displaystyle X_{\text{red}}}
504:{\displaystyle X_{\text{red}}}
434:
416:
402:
344:
259:
207:
157:{\displaystyle {\mathcal {F}}}
110:{\displaystyle {\mathcal {F}}}
1:
1701:
719:we only have to describe the
223:such that for each subscheme
1230:{\displaystyle \mathbb {C} }
216:{\displaystyle i_{Y}:Y\to X}
7:
1108:{\displaystyle 0,1,\infty }
511:is constant as well, where
243:of the covering, the sheaf
164:is called constructible if
10:
1894:
1694:can be computed using the
63:, Exposé IX § 2). For the
1839:10.1007/978-3-662-02541-3
582:is itself constructible.
51:]. It has its origins in
1696:Picard–Lefschetz formula
669:. Since any loop around
606:Derived pushforward on P
1746:Grothendieck, Alexandre
682:{\displaystyle \infty }
596:intersection cohomology
1688:
1662:
1633:
1579:
1438:
1297:
1263:
1231:
1206:
1129:
1109:
1077:
1036:
1012:
980:
948:
916:
889:
757:
737:
713:
683:
663:
576:
552:
532:
505:
478:
455:
389:
369:
318:
298:
237:
217:
178:
158:
131:
111:
49:locally constant sheaf
45:locally closed subsets
1689:
1663:
1634:
1580:
1439:
1298:
1264:
1262:{\displaystyle t=0,1}
1232:
1207:
1130:
1110:
1078:
1037:
1013:
981:
949:
917:
890:
758:
738:
714:
684:
664:
577:
553:
533:
506:
479:
456:
390:
370:
319:
299:
238:
218:
179:
159:
132:
112:
1672:
1643:
1592:
1454:
1310:
1273:
1241:
1219:
1153:
1119:
1087:
1046:
1022:
990:
958:
926:
902:
770:
747:
727:
697:
673:
614:
562:
542:
515:
488:
468:
399:
379:
328:
308:
247:
227:
188:
168:
144:
121:
97:
1801:Sheaves in topology
1765:. pp. vi+640.
1750:Verdier, Jean-Louis
1716:Topics in D-Modules
1687:{\displaystyle 0,1}
712:{\displaystyle 0,1}
286:
22:constructible sheaf
1873:Algebraic geometry
1771:10.1007/BFb0070714
1684:
1658:
1639:are isomorphic to
1629:
1575:
1555:
1493:
1434:
1424:
1399:
1349:
1293:
1259:
1227:
1202:
1125:
1105:
1073:
1032:
1008:
976:
944:
922:are isomorphic to
912:
885:
883:
875:
820:
753:
733:
709:
679:
659:
572:
548:
528:
501:
474:
451:
385:
365:
314:
294:
272:
233:
213:
174:
154:
127:
107:
76:finiteness theorem
53:algebraic geometry
1814:978-3-540-20665-1
1780:978-3-540-06118-2
1546:
1484:
1415:
1390:
1340:
1128:{\displaystyle U}
756:{\displaystyle 1}
736:{\displaystyle 0}
693:to a loop around
551:{\displaystyle X}
525:
498:
477:{\displaystyle X}
388:{\displaystyle Y}
317:{\displaystyle Y}
236:{\displaystyle Y}
177:{\displaystyle X}
130:{\displaystyle X}
34:topological space
1885:
1859:
1825:
1797:Dimca, Alexandru
1792:
1760:
1730:
1728:
1722:, archived from
1721:
1693:
1691:
1690:
1685:
1667:
1665:
1664:
1659:
1657:
1656:
1651:
1638:
1636:
1635:
1630:
1628:
1627:
1608:
1602:
1601:
1584:
1582:
1581:
1576:
1574:
1573:
1556:
1551:
1540:
1539:
1520:
1514:
1513:
1500:
1499:
1494:
1489:
1478:
1477:
1468:
1467:
1462:
1443:
1441:
1440:
1435:
1433:
1432:
1431:
1425:
1420:
1406:
1405:
1400:
1395:
1384:
1383:
1374:
1373:
1368:
1356:
1355:
1350:
1345:
1334:
1333:
1324:
1323:
1318:
1302:
1300:
1299:
1294:
1292:
1268:
1266:
1265:
1260:
1236:
1234:
1233:
1228:
1226:
1211:
1209:
1208:
1203:
1165:
1164:
1134:
1132:
1131:
1126:
1114:
1112:
1111:
1106:
1082:
1080:
1079:
1074:
1072:
1071:
1066:
1041:
1039:
1038:
1033:
1031:
1030:
1017:
1015:
1014:
1009:
1007:
1006:
997:
985:
983:
982:
977:
975:
974:
965:
953:
951:
950:
945:
943:
942:
934:
921:
919:
918:
913:
911:
910:
894:
892:
891:
886:
884:
880:
879:
841:
840:
825:
824:
786:
785:
762:
760:
759:
754:
742:
740:
739:
734:
718:
716:
715:
710:
688:
686:
685:
680:
668:
666:
665:
660:
634:
633:
628:
581:
579:
578:
573:
571:
570:
557:
555:
554:
549:
537:
535:
534:
529:
527:
526:
523:
510:
508:
507:
502:
500:
499:
496:
483:
481:
480:
475:
460:
458:
457:
452:
450:
449:
448:
447:
437:
431:
430:
424:
423:
414:
413:
394:
392:
391:
386:
374:
372:
371:
366:
343:
342:
323:
321:
320:
315:
303:
301:
300:
295:
293:
292:
285:
280:
268:
267:
262:
256:
255:
242:
240:
239:
234:
222:
220:
219:
214:
200:
199:
183:
181:
180:
175:
163:
161:
160:
155:
153:
152:
136:
134:
133:
128:
116:
114:
113:
108:
106:
105:
65:derived category
57:Ă©tale cohomology
1893:
1892:
1888:
1887:
1886:
1884:
1883:
1882:
1863:
1862:
1849:
1815:
1805:Springer-Verlag
1781:
1763:Springer-Verlag
1758:
1752:, eds. (1972).
1737:
1726:
1719:
1709:
1704:
1673:
1670:
1669:
1652:
1647:
1646:
1644:
1641:
1640:
1604:
1603:
1597:
1596:
1595:
1593:
1590:
1589:
1557:
1547:
1545:
1544:
1516:
1515:
1509:
1508:
1507:
1495:
1485:
1483:
1482:
1473:
1469:
1463:
1458:
1457:
1455:
1452:
1451:
1427:
1426:
1416:
1414:
1413:
1401:
1391:
1389:
1388:
1379:
1375:
1369:
1364:
1363:
1351:
1341:
1339:
1338:
1329:
1325:
1319:
1314:
1313:
1311:
1308:
1307:
1288:
1274:
1271:
1270:
1242:
1239:
1238:
1222:
1220:
1217:
1216:
1160:
1156:
1154:
1151:
1150:
1145:elliptic curves
1141:
1120:
1117:
1116:
1088:
1085:
1084:
1067:
1062:
1061:
1047:
1044:
1043:
1026:
1025:
1023:
1020:
1019:
1002:
998:
993:
991:
988:
987:
970:
966:
961:
959:
956:
955:
935:
930:
929:
927:
924:
923:
906:
905:
903:
900:
899:
882:
881:
874:
873:
868:
862:
861:
856:
846:
845:
836:
832:
830:
819:
818:
813:
807:
806:
801:
791:
790:
781:
777:
773:
771:
768:
767:
748:
745:
744:
728:
725:
724:
698:
695:
694:
674:
671:
670:
629:
624:
623:
615:
612:
611:
608:
592:
566:
565:
563:
560:
559:
543:
540:
539:
522:
518:
516:
513:
512:
495:
491:
489:
486:
485:
469:
466:
465:
443:
439:
438:
433:
432:
426:
425:
419:
415:
409:
405:
400:
397:
396:
380:
377:
376:
338:
334:
329:
326:
325:
309:
306:
305:
288:
287:
281:
276:
263:
258:
257:
251:
250:
248:
245:
244:
228:
225:
224:
195:
191:
189:
186:
185:
169:
166:
165:
148:
147:
145:
142:
141:
122:
119:
118:
101:
100:
98:
95:
94:
87:
12:
11:
5:
1891:
1881:
1880:
1875:
1861:
1860:
1847:
1826:
1813:
1793:
1779:
1742:Artin, Michael
1736:
1733:
1732:
1731:
1708:
1705:
1703:
1700:
1683:
1680:
1677:
1655:
1650:
1626:
1623:
1620:
1617:
1614:
1611:
1607:
1600:
1586:
1585:
1572:
1569:
1566:
1563:
1560:
1554:
1550:
1543:
1538:
1535:
1532:
1529:
1526:
1523:
1519:
1512:
1506:
1503:
1498:
1492:
1488:
1481:
1476:
1472:
1466:
1461:
1445:
1444:
1430:
1423:
1419:
1412:
1409:
1404:
1398:
1394:
1387:
1382:
1378:
1372:
1367:
1362:
1359:
1354:
1348:
1344:
1337:
1332:
1328:
1322:
1317:
1291:
1287:
1284:
1281:
1278:
1258:
1255:
1252:
1249:
1246:
1225:
1213:
1212:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1163:
1159:
1140:
1137:
1124:
1104:
1101:
1098:
1095:
1092:
1070:
1065:
1060:
1057:
1054:
1051:
1029:
1005:
1001:
996:
973:
969:
964:
941:
938:
933:
909:
896:
895:
878:
872:
869:
867:
864:
863:
860:
857:
855:
852:
851:
849:
844:
839:
835:
831:
828:
823:
817:
814:
812:
809:
808:
805:
802:
800:
797:
796:
794:
789:
784:
780:
776:
775:
752:
732:
708:
705:
702:
678:
658:
655:
652:
649:
646:
643:
640:
637:
632:
627:
622:
619:
607:
604:
591:
588:
569:
547:
521:
494:
473:
446:
442:
436:
429:
422:
418:
412:
408:
404:
384:
364:
361:
358:
355:
352:
349:
346:
341:
337:
333:
313:
291:
284:
279:
275:
271:
266:
261:
254:
232:
212:
209:
206:
203:
198:
194:
173:
151:
126:
104:
86:
80:
30:abelian groups
9:
6:
4:
3:
2:
1890:
1879:
1876:
1874:
1871:
1870:
1868:
1858:
1854:
1850:
1848:3-540-12175-7
1844:
1840:
1836:
1832:
1827:
1824:
1820:
1816:
1810:
1806:
1802:
1798:
1794:
1790:
1786:
1782:
1776:
1772:
1768:
1764:
1757:
1756:
1751:
1747:
1743:
1739:
1738:
1729:on 2017-09-21
1725:
1718:
1717:
1711:
1710:
1707:Seminar notes
1699:
1697:
1681:
1678:
1675:
1653:
1621:
1618:
1615:
1609:
1567:
1564:
1561:
1552:
1541:
1533:
1530:
1527:
1521:
1504:
1496:
1490:
1474:
1470:
1464:
1450:
1449:
1448:
1421:
1410:
1402:
1396:
1380:
1376:
1370:
1360:
1352:
1346:
1330:
1326:
1320:
1306:
1305:
1304:
1282:
1279:
1276:
1256:
1253:
1250:
1247:
1244:
1196:
1193:
1190:
1181:
1178:
1175:
1169:
1166:
1161:
1157:
1149:
1148:
1147:
1146:
1136:
1122:
1099:
1096:
1093:
1090:
1068:
1055:
1052:
1049:
1003:
999:
971:
967:
939:
936:
876:
870:
865:
858:
853:
847:
842:
837:
833:
826:
821:
815:
810:
803:
798:
792:
787:
782:
778:
766:
765:
764:
750:
730:
722:
706:
703:
700:
692:
650:
647:
644:
641:
635:
630:
620:
617:
603:
601:
597:
587:
583:
545:
519:
492:
471:
462:
444:
440:
420:
410:
406:
382:
359:
356:
353:
350:
347:
339:
335:
311:
282:
277:
273:
269:
264:
230:
210:
204:
201:
196:
192:
171:
138:
124:
92:
91:Ă©tale sheaves
85:
79:
77:
72:
70:
66:
62:
58:
54:
50:
46:
42:
38:
35:
31:
27:
23:
19:
1878:Sheaf theory
1830:
1800:
1754:
1724:the original
1715:
1587:
1446:
1214:
1142:
897:
609:
600:local system
593:
584:
463:
395:, the sheaf
139:
88:
83:
73:
69:â„“-adic sheaf
40:
39:, such that
36:
21:
15:
117:on schemes
55:, where in
18:mathematics
1867:Categories
1735:References
1702:References
32:over some
1610:−
1553:_
1542:⊕
1522:−
1505:≅
1491:_
1475:∗
1471:π
1422:_
1411:≅
1397:_
1381:∗
1377:π
1361:≅
1347:_
1331:∗
1327:π
1286:→
1277:π
1194:−
1179:−
1167:−
1103:∞
1059:→
972:∗
937:⊕
721:monodromy
691:homotopic
677:∞
654:∞
636:−
421:∗
357:∈
351:∣
345:→
283:∗
208:→
1799:(2004),
140:A sheaf
1857:0926276
1823:2050072
1789:0354654
723:around
1855:
1845:
1821:
1811:
1787:
1777:
1759:(PDF)
1727:(PDF)
1720:(PDF)
1303:then
1237:. At
1215:over
26:sheaf
24:is a
1843:ISBN
1809:ISBN
1775:ISBN
1447:and
1042:for
743:and
74:The
20:, a
1835:doi
1767:doi
1018:of
986:or
689:is
524:red
497:red
484:to
28:of
16:In
1869::
1853:MR
1851:,
1841:,
1819:MR
1817:,
1807:,
1785:MR
1783:.
1773:.
1748:;
1744:;
1698:.
1135:.
71:.
1837::
1791:.
1769::
1682:1
1679:,
1676:0
1654:2
1649:Q
1625:}
1622:1
1619:,
1616:0
1613:{
1606:C
1599:L
1571:}
1568:1
1565:,
1562:0
1559:{
1549:Q
1537:}
1534:1
1531:,
1528:0
1525:{
1518:C
1511:L
1502:)
1497:X
1487:Q
1480:(
1465:1
1460:R
1429:C
1418:Q
1408:)
1403:X
1393:Q
1386:(
1371:2
1366:R
1358:)
1353:X
1343:Q
1336:(
1321:0
1316:R
1290:C
1283:X
1280::
1257:1
1254:,
1251:0
1248:=
1245:t
1224:C
1200:)
1197:t
1191:x
1188:(
1185:)
1182:1
1176:x
1173:(
1170:x
1162:2
1158:y
1123:U
1100:,
1097:1
1094:,
1091:0
1069:1
1064:P
1056:U
1053::
1050:j
1028:L
1004:!
1000:j
995:R
968:j
963:R
940:2
932:Q
908:L
877:]
871:1
866:0
859:l
854:1
848:[
843:=
838:1
834:T
827:,
822:]
816:1
811:0
804:k
799:1
793:[
788:=
783:0
779:T
751:1
731:0
707:1
704:,
701:0
657:}
651:,
648:1
645:,
642:0
639:{
631:1
626:P
621:=
618:U
568:F
546:X
520:X
493:X
472:X
445:i
441:U
435:|
428:F
417:)
411:Y
407:i
403:(
383:Y
363:}
360:I
354:i
348:Y
340:i
336:U
332:{
312:Y
290:F
278:Y
274:i
270:=
265:Y
260:|
253:F
231:Y
211:X
205:Y
202::
197:Y
193:i
172:X
150:F
125:X
103:F
84:X
41:X
37:X
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.