768:
159:
220:
1108:
640:
510:
1426:(universally injective). In particular, a closed immersion is universally closed. A closed immersion is stable under base change and composition. The notion of a closed immersion is local in the sense that
973:
1184:
1249:
1712:
435:
344:
823:
1529:
1060:
The only varying condition is the third. It is instructive to look at a counter-example to get a feel for what the third condition yields by looking at a map which is not a closed immersion,
1812:
1319:
1284:
1054:
554:
1388:
1464:
1775:
1023:
997:
694:
1564:
370:
1671:
868:
297:
254:
68:
1620:
1590:
1743:
667:
1408:
1359:
1339:
912:
892:
2009:
703:
93:
2062:
167:
1063:
559:
447:
920:
1116:
1192:
1676:
2088:
2054:
375:
305:
785:
1469:
1780:
2049:
1289:
17:
1254:
1996:
1028:
519:
1364:
1433:
32:
1756:
1004:
978:
675:
36:
1817:
A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.
1593:
1537:
24:
349:
1341:
the sheaf has no sections. This violates the third condition since at least one open subscheme
2058:
2000:
1831:
1644:
1423:
841:
270:
225:
41:
2044:
2018:
1599:
1569:
79:
2072:
2030:
1721:
645:
2068:
2026:
1826:
1419:
2038:
1940:
1393:
1344:
1324:
897:
877:
2082:
1877:"Section 26.4 (01HJ): Closed immersions of locally ringed spaces—The Stacks project"
763:{\displaystyle f_{\ast }{\mathcal {O}}_{Z}\cong {\mathcal {O}}_{X}/{\mathcal {I}}}
2004:
1430:
is a closed immersion if and only if for some (equivalently every) open covering
779:
154:{\displaystyle f^{\#}:{\mathcal {O}}_{X}\rightarrow f_{\ast }{\mathcal {O}}_{Z}}
1901:"Section 17.8 (01B1): Modules locally generated by sections—The Stacks project"
1936:
1900:
1876:
2022:
215:{\displaystyle \operatorname {Spec} (R/I)\to \operatorname {Spec} (R)}
1937:"Part 4: Algebraic Spaces, Chapter 67: Morphisms of Algebraic Spaces"
1103:{\displaystyle i:\mathbb {G} _{m}\hookrightarrow \mathbb {A} ^{1}}
870:
is a closed immersion if a similar list of criteria is satisfied
1753:
is exact, fully faithful with the essential image consisting of
1286:
then there are no sections. This implies for any open subscheme
635:{\displaystyle f^{-1}(U_{j})=\operatorname {Spec} (R_{j}/I_{j})}
505:{\displaystyle X=\bigcup U_{j},U_{j}=\operatorname {Spec} R_{j}}
2005:"Éléments de géométrie algébrique: I. Le langage des schémas"
968:{\displaystyle {\mathcal {O}}_{X}\to i_{*}{\mathcal {O}}_{Z}}
1793:
1786:
1762:
1693:
1682:
1209:
1179:{\displaystyle \mathbb {G} _{m}={\text{Spec}}(\mathbb {Z} )}
1035:
1010:
984:
954:
927:
810:
792:
755:
737:
720:
681:
140:
113:
1244:{\displaystyle i_{*}{\mathcal {O}}_{\mathbb {G} _{m}}|_{0}}
1707:{\displaystyle {\mathcal {I}}\subset {\mathcal {O}}_{X}}
90:. The latter condition can be formalized by saying that
1961:
1923:
1783:
1759:
1724:
1679:
1647:
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1540:
1472:
1436:
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1347:
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1007:
981:
923:
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844:
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706:
678:
648:
562:
522:
450:
378:
352:
308:
273:
228:
170:
96:
44:
430:{\displaystyle f^{-1}(U)=\operatorname {Spec} (R/I)}
339:{\displaystyle U=\operatorname {Spec} (R)\subset X}
1806:
1769:
1737:
1714:is the quasi-coherent sheaf of ideals cutting out
1706:
1665:
1614:
1584:
1558:
1523:
1458:
1402:
1382:
1353:
1333:
1313:
1278:
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1178:
1102:
1048:
1017:
991:
967:
906:
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862:
817:
762:
688:
661:
634:
548:
504:
429:
364:
338:
291:
248:
214:
153:
62:
1745:from the category of quasi-coherent sheaves over
818:{\displaystyle {\mathcal {O}}_{X}/{\mathcal {I}}}
1524:{\displaystyle f:f^{-1}(U_{j})\rightarrow U_{j}}
838:In the case of locally ringed spaces a morphism
1749:to the category of quasi-coherent sheaves over
1807:{\displaystyle {\mathcal {I}}{\mathcal {G}}=0}
16:For the concept in differential geometry, see
8:
2057:, vol. 52, New York: Springer-Verlag,
1863:
672:There is a quasi-coherent sheaf of ideals
1792:
1791:
1785:
1784:
1782:
1761:
1760:
1758:
1729:
1723:
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1681:
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1499:
1483:
1471:
1450:
1435:
1395:
1374:
1370:
1369:
1366:
1346:
1326:
1314:{\displaystyle U\subset \mathbb {A} ^{1}}
1305:
1301:
1300:
1291:
1270:
1266:
1265:
1256:
1235:
1230:
1221:
1217:
1216:
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1207:
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1194:
1161:
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1009:
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1006:
983:
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945:
932:
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922:
899:
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748:
742:
736:
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719:
718:
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680:
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677:
653:
647:
623:
614:
608:
583:
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561:
540:
527:
521:
496:
477:
464:
449:
416:
383:
377:
351:
307:
272:
238:
227:
183:
169:
145:
139:
138:
131:
118:
112:
111:
101:
95:
43:
1982:Stacks, Morphisms of schemes. Lemma 27.2
1973:Stacks, Morphisms of schemes. Lemma 4.1
1843:
1025:is locally generated by sections as an
1852:The Red Book of Varieties and Schemes
1279:{\displaystyle 0\in \mathbb {A} ^{1}}
444:There exists an open affine covering
7:
2010:Publications Mathématiques de l'IHÉS
834:Definition for locally ringed spaces
1049:{\displaystyle {\mathcal {O}}_{X}}
549:{\displaystyle I_{j}\subset R_{j}}
102:
14:
1962:Grothendieck & Dieudonné 1960
1924:Grothendieck & Dieudonné 1960
1383:{\displaystyle \mathbb {A} ^{1}}
164:An example is the inclusion map
1459:{\displaystyle X=\bigcup U_{j}}
1770:{\displaystyle {\mathcal {G}}}
1657:
1606:
1576:
1550:
1544:
1508:
1505:
1492:
1231:
1173:
1170:
1148:
1140:
1085:
1018:{\displaystyle {\mathcal {I}}}
992:{\displaystyle {\mathcal {I}}}
938:
854:
689:{\displaystyle {\mathcal {I}}}
629:
601:
589:
576:
424:
410:
398:
392:
327:
321:
283:
264:The following are equivalent:
232:
209:
203:
194:
191:
177:
124:
54:
1:
2055:Graduate Texts in Mathematics
222:induced by the canonical map
1559:{\displaystyle Z\to Y\to X}
1189:If we look at the stalk of
2105:
1673:is a closed immersion and
1622:is a closed immersion. If
1566:is a closed immersion and
975:is surjective with kernel
365:{\displaystyle I\subset R}
15:
917:The associated sheaf map
1905:stacks.math.columbia.edu
1881:stacks.math.columbia.edu
1718:, then the direct image
1666:{\displaystyle i:Z\to X}
863:{\displaystyle i:Z\to X}
346:, there exists an ideal
292:{\displaystyle f:Z\to X}
249:{\displaystyle R\to R/I}
63:{\displaystyle f:Z\to X}
1997:Grothendieck, Alexandre
1638:is a closed immersion.
1531:is a closed immersion.
260:Other characterizations
18:Immersion (mathematics)
1808:
1771:
1739:
1708:
1667:
1616:
1615:{\displaystyle Z\to Y}
1586:
1585:{\displaystyle Y\to X}
1560:
1525:
1460:
1418:A closed immersion is
1404:
1384:
1355:
1335:
1315:
1280:
1245:
1187:
1180:
1104:
1050:
1019:
993:
969:
908:
894:is a homeomorphism of
888:
864:
819:
764:
690:
663:
636:
550:
516:there exists an ideal
506:
431:
366:
340:
302:For every open affine
299:is a closed immersion.
293:
250:
216:
155:
74:as a closed subset of
64:
1945:, Columbia University
1809:
1772:
1740:
1738:{\displaystyle i_{*}}
1709:
1668:
1617:
1587:
1561:
1526:
1461:
1405:
1385:
1356:
1336:
1316:
1281:
1246:
1181:
1112:
1105:
1051:
1020:
994:
970:
909:
889:
865:
820:
774:is an isomorphism of
765:
691:
664:
662:{\displaystyle U_{j}}
637:
551:
507:
432:
367:
341:
294:
251:
217:
156:
65:
2089:Morphisms of schemes
1781:
1757:
1722:
1677:
1645:
1630:-scheme, then every
1600:
1570:
1538:
1470:
1434:
1394:
1365:
1345:
1325:
1290:
1255:
1193:
1117:
1064:
1029:
1005:
979:
921:
898:
878:
842:
786:
704:
676:
646:
560:
520:
448:
376:
350:
306:
271:
226:
168:
94:
42:
1534:If the composition
86:can be extended to
78:such that locally,
37:morphism of schemes
2050:Algebraic Geometry
2023:10.1007/bf02684778
1942:The stacks project
1804:
1767:
1735:
1704:
1663:
1612:
1582:
1556:
1521:
1456:
1400:
1380:
1351:
1331:
1311:
1276:
1241:
1176:
1100:
1046:
1015:
989:
965:
904:
884:
860:
815:
760:
686:
659:
632:
546:
502:
427:
362:
336:
289:
246:
212:
151:
60:
25:algebraic geometry
2064:978-0-387-90244-9
2045:Hartshorne, Robin
1832:Regular embedding
1403:{\displaystyle 0}
1354:{\displaystyle U}
1334:{\displaystyle 0}
1138:
907:{\displaystyle Z}
887:{\displaystyle i}
80:regular functions
2096:
2075:
2034:
1983:
1980:
1974:
1971:
1965:
1959:
1953:
1952:
1951:
1950:
1933:
1927:
1921:
1915:
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1912:
1911:
1897:
1891:
1890:
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1873:
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1861:
1855:
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1813:
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1805:
1797:
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1776:
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1773:
1768:
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1744:
1742:
1741:
1736:
1734:
1733:
1713:
1711:
1710:
1705:
1703:
1702:
1697:
1696:
1686:
1685:
1672:
1670:
1669:
1664:
1621:
1619:
1618:
1613:
1591:
1589:
1588:
1583:
1565:
1563:
1562:
1557:
1530:
1528:
1527:
1522:
1520:
1519:
1504:
1503:
1491:
1490:
1466:the induced map
1465:
1463:
1462:
1457:
1455:
1454:
1409:
1407:
1406:
1401:
1389:
1387:
1386:
1381:
1379:
1378:
1373:
1360:
1358:
1357:
1352:
1340:
1338:
1337:
1332:
1320:
1318:
1317:
1312:
1310:
1309:
1304:
1285:
1283:
1282:
1277:
1275:
1274:
1269:
1250:
1248:
1247:
1242:
1240:
1239:
1234:
1228:
1227:
1226:
1225:
1220:
1213:
1212:
1205:
1204:
1185:
1183:
1182:
1177:
1169:
1168:
1147:
1139:
1136:
1131:
1130:
1125:
1109:
1107:
1106:
1101:
1099:
1098:
1093:
1084:
1083:
1078:
1055:
1053:
1052:
1047:
1045:
1044:
1039:
1038:
1024:
1022:
1021:
1016:
1014:
1013:
998:
996:
995:
990:
988:
987:
974:
972:
971:
966:
964:
963:
958:
957:
950:
949:
937:
936:
931:
930:
913:
911:
910:
905:
893:
891:
890:
885:
869:
867:
866:
861:
824:
822:
821:
816:
814:
813:
807:
802:
801:
796:
795:
769:
767:
766:
761:
759:
758:
752:
747:
746:
741:
740:
730:
729:
724:
723:
716:
715:
695:
693:
692:
687:
685:
684:
668:
666:
665:
660:
658:
657:
642:as schemes over
641:
639:
638:
633:
628:
627:
618:
613:
612:
588:
587:
575:
574:
555:
553:
552:
547:
545:
544:
532:
531:
511:
509:
508:
503:
501:
500:
482:
481:
469:
468:
437:as schemes over
436:
434:
433:
428:
420:
391:
390:
371:
369:
368:
363:
345:
343:
342:
337:
298:
296:
295:
290:
255:
253:
252:
247:
242:
221:
219:
218:
213:
187:
160:
158:
157:
152:
150:
149:
144:
143:
136:
135:
123:
122:
117:
116:
106:
105:
70:that identifies
69:
67:
66:
61:
29:closed immersion
2104:
2103:
2099:
2098:
2097:
2095:
2094:
2093:
2079:
2078:
2065:
2043:
2001:Dieudonné, Jean
1995:
1992:
1987:
1986:
1981:
1977:
1972:
1968:
1960:
1956:
1948:
1946:
1935:
1934:
1930:
1922:
1918:
1909:
1907:
1899:
1898:
1894:
1885:
1883:
1875:
1874:
1870:
1864:Hartshorne 1977
1862:
1858:
1849:
1845:
1840:
1827:Segre embedding
1823:
1779:
1778:
1755:
1754:
1725:
1720:
1719:
1690:
1675:
1674:
1643:
1642:
1626:is a separated
1598:
1597:
1568:
1567:
1536:
1535:
1511:
1495:
1479:
1468:
1467:
1446:
1432:
1431:
1416:
1392:
1391:
1368:
1363:
1362:
1343:
1342:
1323:
1322:
1299:
1288:
1287:
1264:
1253:
1252:
1229:
1215:
1206:
1196:
1191:
1190:
1157:
1120:
1115:
1114:
1088:
1073:
1062:
1061:
1032:
1027:
1026:
1003:
1002:
977:
976:
951:
941:
924:
919:
918:
896:
895:
876:
875:
840:
839:
836:
789:
784:
783:
734:
717:
707:
702:
701:
674:
673:
649:
644:
643:
619:
604:
579:
563:
558:
557:
536:
523:
518:
517:
492:
473:
460:
446:
445:
379:
374:
373:
348:
347:
304:
303:
269:
268:
262:
224:
223:
166:
165:
161:is surjective.
137:
127:
110:
97:
92:
91:
40:
39:
21:
12:
11:
5:
2102:
2100:
2092:
2091:
2081:
2080:
2077:
2076:
2063:
2041:
2039:Stacks Project
2035:
1991:
1988:
1985:
1984:
1975:
1966:
1954:
1928:
1916:
1892:
1868:
1856:
1854:, Section II.5
1842:
1841:
1839:
1836:
1835:
1834:
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1822:
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1803:
1800:
1795:
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1701:
1695:
1689:
1684:
1662:
1659:
1656:
1653:
1650:
1611:
1608:
1605:
1581:
1578:
1575:
1555:
1552:
1549:
1546:
1543:
1518:
1514:
1510:
1507:
1502:
1498:
1494:
1489:
1486:
1482:
1478:
1475:
1453:
1449:
1445:
1442:
1439:
1415:
1412:
1399:
1377:
1372:
1350:
1330:
1308:
1303:
1298:
1295:
1273:
1268:
1263:
1260:
1238:
1233:
1224:
1219:
1211:
1203:
1199:
1175:
1172:
1167:
1164:
1160:
1156:
1153:
1150:
1146:
1142:
1134:
1129:
1124:
1097:
1092:
1087:
1082:
1077:
1072:
1069:
1058:
1057:
1043:
1037:
1012:
999:
986:
962:
956:
948:
944:
940:
935:
929:
915:
914:onto its image
903:
883:
859:
856:
853:
850:
847:
835:
832:
831:
830:
812:
806:
800:
794:
757:
751:
745:
739:
733:
728:
722:
714:
710:
683:
670:
656:
652:
631:
626:
622:
617:
611:
607:
603:
600:
597:
594:
591:
586:
582:
578:
573:
570:
566:
543:
539:
535:
530:
526:
499:
495:
491:
488:
485:
480:
476:
472:
467:
463:
459:
456:
453:
442:
426:
423:
419:
415:
412:
409:
406:
403:
400:
397:
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389:
386:
382:
361:
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320:
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288:
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245:
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211:
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199:
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148:
142:
134:
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126:
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115:
109:
104:
100:
59:
56:
53:
50:
47:
13:
10:
9:
6:
4:
3:
2:
2101:
2090:
2087:
2086:
2084:
2074:
2070:
2066:
2060:
2056:
2052:
2051:
2046:
2042:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2012:
2011:
2006:
2002:
1998:
1994:
1993:
1989:
1979:
1976:
1970:
1967:
1963:
1958:
1955:
1944:
1943:
1938:
1932:
1929:
1925:
1920:
1917:
1906:
1902:
1896:
1893:
1882:
1878:
1872:
1869:
1865:
1860:
1857:
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1820:
1818:
1815:
1801:
1798:
1752:
1748:
1730:
1726:
1717:
1699:
1687:
1660:
1654:
1651:
1648:
1639:
1637:
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1634:-section of
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1001:The kernel
780:global Spec
1990:References
1949:2024-03-06
1910:2021-08-05
1886:2021-08-05
1777:such that
1414:Properties
700:such that
556:such that
372:such that
1850:Mumford,
1731:∗
1688:⊂
1658:→
1607:→
1594:separated
1577:→
1551:→
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1485:−
1444:⋃
1390:contains
1361:covering
1297:⊂
1262:∈
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1086:↪
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778:onto the
732:≅
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103:#
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2083:Category
2047:(1977),
2003:(1960).
1821:See also
1424:radicial
874:The map
2073:0463157
2031:0217083
1964:, 5.4.6
1926:, 4.2.4
1866:, §II.3
1596:, then
1056:-module
33:schemes
2071:
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2029:
1420:finite
1838:Notes
1110:where
825:over
35:is a
2059:ISBN
2037:The
1422:and
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770:and
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487:Spec
405:Spec
316:Spec
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172:Spec
27:, a
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