1109:
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944:
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1207:
553:
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Forming sheaf categories and direct image functors itself defines a functor from the category of topological spaces to the category of categories: given continuous maps
1551:
910:
1683:
1264:
1740:
1760:
1104:{\displaystyle \mathrm {Hom} _{\mathbf {Sh} (X)}(f^{-1}{\mathcal {G}},{\mathcal {F}})=\mathrm {Hom} _{\mathbf {Sh} (Y)}({\mathcal {G}},f_{*}{\mathcal {F}})}
343:
1887:
1855:
383:
143:, which is known as the direct image functor. Similar constructions exist in many other algebraic and geometric contexts, including that of
1713:
780:
preserves the property of being quasi-coherent, so we obtain the direct image functor between categories of quasi-coherent sheaves.
204:
191:
336:
1284:
1559:
260:
1775:
915:
329:
161:
1442:
1147:
688:
If dealing with sheaves of sets instead of sheaves of abelian groups, the same definition applies. Similarly, if
216:
1165:
770:
482:
677:
1377:
766:
144:
132:
1882:
1212:
877:
181:
319:
152:
574:
1340:
1688:
1652:
48:
40:
1417:
One can show that there is a similar expression as above for higher direct images: for a sheaf
1851:
1524:
883:
788:
56:
1658:
1515:
1390:
1243:
1865:
1718:
373:
be a continuous map of topological spaces, and let Sh(–) denote the category of sheaves of
1861:
1847:
1763:
1400:
873:
1553:
of quasi-compact and quasi-separated schemes, one likewise has the right derived functor
1745:
148:
63:
17:
1876:
1396:
1136:
800:
374:
1796:"Section 26.24 (01LA): Functoriality for quasi-coherent modules—The Stacks project"
715:
36:
28:
1819:
1795:
44:
122:
1820:"Section 48.3 (0A9D): Right adjoint of pushforward—The Stacks project"
436:{\displaystyle f_{*}:\operatorname {Sh} (X)\to \operatorname {Sh} (Y)}
784:
1712:. This is closely related, but not generally equivalent to, the
1399:, but usually not right exact. Hence one can consider the right
562:, and is called the direct image sheaf or pushforward sheaf of
1219:
1184:
1093:
1073:
1025:
1015:
931:
921:
43:
to the relative case. It is of fundamental importance in
1330:{\displaystyle \mathrm {Spec} \,S\to \mathrm {Spec} \,R}
1748:
1721:
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1168:
964:
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386:
263:
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1521:
In the context of algebraic geometry and a morphism
377:on a topological space. The direct image functor
1754:
1734:
1704:
1677:
1640:
1545:
1498:
1361:
1329:
1258:
1232:
1201:
1103:
938:
904:
547:
435:
307:
245:
1641:{\displaystyle Rf_{*}:D_{qcoh}(X)\to D_{qcoh}(Y)}
308:{\displaystyle (R)f_{!}\leftrightarrows (R)f^{!}}
661:of abelian groups, and the direct image functor
1162:. This follows from the fact that the stalk of
1655:of quasi-coherent sheaves. In this situation,
1376:on quasi-coherent sheaves identifies with the
939:{\displaystyle {\mathcal {F}},{\mathcal {G}}}
337:
8:
1499:{\displaystyle U\mapsto H^{q}(f^{-1}(U),F)}
1266:and zero otherwise (here the closedness of
783:A similar definition applies to sheaves on
246:{\displaystyle f^{*}\leftrightarrows f_{*}}
1436:) is the sheaf associated to the presheaf
344:
330:
157:
1747:
1726:
1720:
1696:
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1669:
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1614:
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1202:{\displaystyle (f_{*}{\mathcal {F}})_{y}}
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391:
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548:{\displaystyle f_{*}F(U):=F(f^{-1}(U)).}
1787:
791:. There, instead of the above preimage
754:-modules to the category of sheaves of
622:in an obvious way, we indeed have that
160:
1120:is the inclusion of a closed subspace
880:, which means that for any continuous
1651:as a functor between the (unbounded)
1403:of the direct image. They are called
7:
589:gives rise to a morphism of sheaves
1150:between the category of sheaves on
718:, we obtain a direct image functor
653:the unique continuous map, then Sh(
117:is given by the global sections of
107:, such that the global sections of
1846:, Universitext, Berlin, New York:
1337:determined by a ring homomorphism
1319:
1316:
1313:
1310:
1298:
1295:
1292:
1289:
1281:is the morphism of affine schemes
1233:{\displaystyle {\mathcal {F}}_{y}}
1044:
1041:
1038:
973:
970:
967:
747:) from the category of sheaves of
192:direct image with compact support
121:. This assignment gives rise to a
25:
1714:exceptional inverse image functor
954:, there is a natural isomorphism:
1369:, then the direct image functor
1053:
1050:
982:
979:
558:This turns out to be a sheaf on
1154:and the category of sheaves on
1888:Theory of continuous functions
1685:always admits a right adjoint
1635:
1629:
1607:
1604:
1598:
1537:
1493:
1484:
1478:
1462:
1449:
1353:
1306:
1190:
1169:
1098:
1068:
1063:
1057:
1030:
997:
992:
986:
896:
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415:
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292:
286:
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139:to the category of sheaves on
1:
454:to its direct image presheaf
1395:The direct image functor is
1362:{\displaystyle \phi :R\to S}
872:The direct image functor is
77:, we can define a new sheaf
1705:{\displaystyle f^{\times }}
1904:
1776:Proper base change theorem
1388:
468:, defined on open subsets
205:exceptional inverse image
162:Image functors for sheaves
1139:. Actually, in this case
1842:Iversen, Birger (1986),
1824:stacks.math.columbia.edu
1800:stacks.math.columbia.edu
1546:{\displaystyle f:X\to Y}
905:{\displaystyle f:X\to Y}
761:-modules. Moreover, if
678:global sections functor
41:global sections functor
18:Direct image of a sheaf
1756:
1736:
1706:
1679:
1678:{\displaystyle Rf_{*}}
1642:
1547:
1500:
1378:restriction of scalars
1363:
1331:
1260:
1259:{\displaystyle y\in X}
1234:
1203:
1105:
940:
906:
549:
437:
309:
247:
145:quasi-coherent sheaves
1844:Cohomology of sheaves
1757:
1737:
1735:{\displaystyle f^{!}}
1707:
1680:
1643:
1548:
1501:
1364:
1332:
1261:
1235:
1204:
1106:
941:
907:
878:inverse image functor
765:is now a morphism of
550:
438:
310:
248:
39:that generalizes the
35:is a construction in
1746:
1719:
1689:
1659:
1560:
1525:
1443:
1405:higher direct images
1385:Higher direct images
1341:
1285:
1244:
1213:
1166:
962:
916:
884:
483:
384:
320:Base change theorems
261:
217:
33:direct image functor
1869:, esp. section II.4
714:) is a morphism of
575:morphism of sheaves
1752:
1732:
1702:
1675:
1653:derived categories
1638:
1543:
1496:
1359:
1327:
1256:
1230:
1199:
1101:
936:
902:
657:) is the category
545:
433:
305:
243:
93:direct image sheaf
49:algebraic geometry
1857:978-3-540-16389-3
1755:{\displaystyle f}
354:
353:
97:pushforward sheaf
57:topological space
16:(Redirected from
1895:
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1711:
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1700:
1684:
1682:
1681:
1676:
1674:
1673:
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1639:
1628:
1627:
1597:
1596:
1575:
1574:
1552:
1550:
1549:
1544:
1516:sheaf cohomology
1505:
1503:
1502:
1497:
1477:
1476:
1461:
1460:
1401:derived functors
1391:Coherent duality
1380:functor along φ.
1368:
1366:
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1019:
1018:
1012:
1011:
996:
995:
985:
976:
946:respectively on
945:
943:
942:
937:
935:
934:
925:
924:
911:
909:
908:
903:
799:), one uses the
641:is a point, and
554:
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528:
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51:. Given a sheaf
21:
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1848:Springer-Verlag
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1665:
1657:
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1566:
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1523:
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1452:
1441:
1440:
1431:
1413:
1393:
1387:
1375:
1339:
1338:
1283:
1282:
1242:
1241:
1216:
1211:
1210:
1189:
1172:
1164:
1163:
1145:
1134:
1081:
1036:
1000:
965:
960:
959:
914:
913:
882:
881:
868:
862:
855:
821:
779:
771:quasi-separated
759:
752:
745:
734:
724:
712:
701:
686:
667:
635:
628:
613:
602:
595:
517:
486:
481:
480:
460:
387:
382:
381:
359:
350:
295:
273:
259:
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233:
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130:
113:
83:
23:
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15:
12:
11:
5:
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1778:
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1768:
1751:
1729:
1725:
1699:
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1672:
1668:
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1600:
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1472:
1468:
1464:
1459:
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1386:
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1297:
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898:
895:
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777:
773:schemes, then
757:
750:
743:
732:
722:
710:
699:
685:
682:
665:
634:
631:
629:is a functor.
626:
611:
600:
593:
556:
555:
544:
541:
538:
535:
532:
527:
524:
520:
516:
513:
510:
507:
504:
501:
498:
493:
489:
458:
446:sends a sheaf
444:
443:
432:
429:
426:
423:
420:
417:
414:
411:
408:
405:
402:
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394:
390:
375:abelian groups
358:
355:
352:
351:
349:
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302:
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266:
255:
254:
240:
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223:
211:
210:
201:
200:
196:
188:
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182:inverse image
178:
177:
173:
165:
164:
135:of sheaves on
128:
111:
81:
64:continuous map
24:
14:
13:
10:
9:
6:
4:
3:
2:
1900:
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1769:
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1554:
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1531:
1528:
1519:
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1513:
1490:
1487:
1481:
1473:
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1466:
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1453:
1446:
1439:
1438:
1437:
1435:
1428:
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1402:
1398:
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1356:
1350:
1347:
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1324:
1303:
1280:
1276:
1273:
1269:
1253:
1250:
1247:
1225:
1194:
1177:
1173:
1161:
1158:supported on
1157:
1153:
1149:
1142:
1138:
1131:
1127:
1123:
1119:
1115:
1114:
1086:
1082:
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1033:
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1008:
1005:
1001:
989:
958:
957:
953:
949:
926:
899:
893:
890:
887:
879:
875:
874:right adjoint
871:
865:
859:
851:
847:
843:
839:
835:
831:
827:
823:
822:
818:
816:
814:
810:
806:
802:
801:fiber product
798:
794:
790:
789:étale sheaves
786:
781:
776:
772:
768:
767:quasi-compact
764:
760:
753:
746:
739:
735:
728:
721:
717:
716:ringed spaces
713:
706:
702:
695:
691:
683:
681:
679:
675:
671:
664:
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648:
644:
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632:
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621:
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606:
599:
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584:
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571:
569:
565:
561:
542:
533:
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518:
511:
508:
502:
496:
491:
487:
479:
478:
477:
475:
471:
467:
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457:
453:
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427:
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409:
403:
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388:
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378:
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372:
368:
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342:
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328:
327:
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324:
321:
318:
317:
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278:
274:
267:
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238:
234:
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213:
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202:
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186:
185:
180:
179:
176:
172:
169:direct image
167:
166:
163:
159:
156:
154:
150:
149:étale sheaves
146:
142:
138:
134:
127:
124:
120:
116:
110:
106:
102:
98:
94:
91:, called the
90:
86:
80:
76:
72:
68:
65:
61:
58:
55:defined on a
54:
50:
46:
42:
38:
34:
30:
19:
1883:Sheaf theory
1843:
1827:. Retrieved
1823:
1814:
1803:. Retrieved
1799:
1790:
1650:
1520:
1511:
1509:
1433:
1426:
1425:, the sheaf
1422:
1418:
1416:
1408:
1407:and denoted
1404:
1394:
1370:
1278:
1271:
1267:
1159:
1155:
1151:
1140:
1129:
1125:
1121:
1117:
951:
947:
912:and sheaves
863:
857:
849:
845:
841:
837:
833:
829:
825:
812:
808:
804:
796:
792:
782:
774:
762:
755:
748:
741:
737:
730:
726:
719:
708:
704:
697:
693:
689:
687:
673:
669:
662:
658:
654:
650:
646:
642:
638:
636:
623:
619:
615:
608:
604:
597:
590:
586:
582:
578:
572:
567:
563:
559:
557:
473:
469:
465:
461:
455:
451:
447:
445:
370:
366:
362:
360:
206:
193:
183:
170:
168:
140:
136:
125:
118:
114:
108:
104:
100:
96:
92:
88:
84:
78:
74:
70:
66:
59:
52:
37:sheaf theory
32:
26:
1148:equivalence
848:, we have (
676:equals the
29:mathematics
1877:Categories
1829:2022-09-20
1805:2022-09-20
1782:References
1397:left exact
1389:See also:
819:Properties
787:, such as
357:Definition
1742:, unless
1698:×
1671:∗
1608:→
1572:∗
1538:→
1471:−
1450:↦
1354:→
1345:ϕ
1307:→
1274:is used).
1251:∈
1178:∗
1087:∗
1006:−
897:→
523:−
492:∗
422:
416:→
404:
393:∗
284:⇆
239:∗
231:⇆
226:∗
131:from the
1770:See also
1762:is also
1514:denotes
684:Variants
573:Since a
133:category
45:topology
1866:0842190
876:to the
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