1136:
1008:
602:. In the second edition of EGA constructible sets (according to the definition above) are called "globally constructible" while the word "constructible" is reserved for what are called locally constructible above.
572:
subsets are retrocompact, and so for such spaces the simplified definition given first above is equivalent to the more elaborate one. Most of the commonly met schemes in algebraic geometry (including all
776:
59:). In addition, a large number of "local" geometric properties of schemes, morphisms and sheaves are (locally) constructible. Constructible sets also feature in the definition of various types of
1634:
899:
1943:
1753:
1509:
1318:
1171:
954:
561:
and finite unions (and hence also finite intersections) of sets in it. In other words, constructible sets are precisely the
Boolean algebra generated by retrocompact open subsets.
1283:
1226:
1039:
920:
on schemes hold true over a locally constructible subset. EGA IV § 9 covers a large number of such properties. Below are some examples (where all references point to EGA IV):
408:
1415:
535:
1195:
2107:
2065:
2043:
1908:
1866:
1718:
1676:
461:
265:
1969:
1779:
1555:
1344:
680:
355:
199:
1844:
1454:
1070:
650:
169:
1995:
1805:
1581:
1370:
1252:
1065:
843:
808:
488:
709:
595:
2085:
1886:
1696:
1654:
1529:
1477:
729:
555:
511:
428:
329:
305:
285:
219:
139:
119:
2515:
2473:
2431:
959:
734:
In particular, the image of an algebraic variety need not be a variety, but is (under the assumptions) always a constructible set. For example, the map
1720:
is any of the following properties: surjective, proper, finite, immersion, closed immersion, open immersion, isomorphism. (Proposition (9.6.1))
2564:
565:
95:.) However, a modification and another slightly weaker definition are needed to have definitions that behave better with "large" spaces:
904:
Chevalley's theorem in the generality stated above would fail if the simplified definition of constructible sets (without restricting to
2391:
2556:
737:
2113:
One important role that these constructibility results have is that in most cases assuming the morphisms in questions are also
1910:
is any of the following properties: geometrically irreducible, geometrically connected, geometrically reduced. (Theorem (9.7.7))
614:
of a (locally) constructible set is also (locally) constructible for a large class of maps (or "morphisms"). The key result is:
2382:. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) --- Results in Mathematics and Related Areas (3). Vol. 33. Berlin:
2109:
is any of the following properties: geometrically regular, geometrically normal, geometrically reduced. (Proposition (9.9.4))
653:
2582:
1586:
2631:
84:
2581:. Studies in Logic and the Foundations of Mathematics. Amsterdam --- Warsaw: North-Holland Publishing Co. ----
2511:"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie"
848:
2469:"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie"
2555:. Grundlehren der Mathematischen Wissenschaften (in French). Vol. 166 (2nd ed.). Berlin; New York:
43:
in algebraic geometry shows that the image of a constructible set is constructible for an important class of
1916:
1726:
1482:
1291:
1144:
927:
558:
64:
2370:
2544:
2502:
2460:
2418:
2130:
17:
1257:
1200:
1013:
367:
2427:"Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie"
1375:
516:
56:
1176:
2135:
917:
611:
577:) are locally Noetherian, but there are important constructions that lead to more general schemes.
60:
52:
48:
2090:
2048:
2000:
1891:
1849:
1701:
1659:
1131:{\displaystyle {\mathcal {F}}'_{s}\rightarrow {\mathcal {F}}_{s}\rightarrow {\mathcal {F}}''_{s}}
433:
232:
80:
76:
36:
1948:
1758:
1534:
1323:
659:
334:
178:
1810:
1420:
2626:
2574:
2560:
2548:
2506:
2464:
2422:
2387:
623:
574:
44:
32:
148:
2524:
2482:
2440:
1974:
1784:
1560:
1349:
1231:
1044:
813:
581:
362:
2590:
2536:
2494:
2452:
2401:
781:
466:
2586:
2532:
2490:
2448:
2397:
2383:
685:
610:
A major reason for the importance of constructible sets in algebraic geometry is that the
75:
A simple definition, adequate in many situations, is that a constructible set is a finite
2070:
1871:
1681:
1639:
1514:
1462:
714:
599:
540:
496:
413:
314:
290:
270:
204:
124:
104:
2510:
2620:
2114:
1003:{\displaystyle {\mathcal {F}}'\rightarrow {\mathcal {F}}\rightarrow {\mathcal {F}}''}
172:
2408:
2468:
2426:
2355:, § 12 Étude des fibres des morphismes plats de présentation finie, pp. 173-187.
2613:
Constructibility properties of morphisms of schemes (incl. Chevalley's theorem)
2217:
92:
2610:
2604:
2309:
2269:
2174:
1945:
be an locally finitely presented morphism of schemes and consider the set
585:
2121:
subset. A substantial number of such results is included in EGA IV § 12.
88:
35:
that have a relatively "simple" structure. They are used particularly in
24:
2218:"Rigid geometric structures, isometric actions, and algebraic quotients"
2528:
2486:
2444:
557:
that (i) contains all open retrocompact subsets and (ii) contains all
1755:
be an finitely presented morphism of schemes and consider the set
594:
The definition given here is the one used by the first edition of
916:
A large number of "local" properties of morphisms of schemes and
2378:
Andradas, Carlos; Bröcker, Ludwig; Ruiz, Jesús M. (1996).
1285:
is locally free is locally constructible. (Proposition (9.4.7))
2117:
it follows that the properties in question in fact hold in an
493:
Equivalently the constructible subsets of a topological space
2310:"Section 109.24 Images of locally closed subsets (tag 0GZL)"
1264:
1207:
1182:
1114:
1097:
1077:
1020:
991:
980:
966:
771:{\displaystyle \mathbf {A} ^{2}\rightarrow \mathbf {A} ^{2}}
2553:Éléments de géométrie algébrique: I. Le langage des schémas
1138:
is exact is locally constructible. (Proposition (9.4.4))
584:) topological space, every constructible set contains a
430:
consisting of open subsets with the property that each
2243:, DĂ©finitions (2.3.1), (2.3.2) and (2.3.10), pp. 55-57
2164:, DĂ©finitions (9.1.1), (9.1.2) and (9.1.11), pp. 12-14
2371:
Note on the constructible sets of a topological space
2348:
2332:
2292:
2252:
2233:
2197:
2154:
2093:
2073:
2051:
2003:
1977:
1951:
1919:
1894:
1874:
1852:
1813:
1787:
1761:
1729:
1704:
1684:
1662:
1642:
1589:
1563:
1537:
1517:
1485:
1465:
1423:
1378:
1352:
1326:
1294:
1260:
1234:
1203:
1179:
1147:
1073:
1047:
1016:
962:
930:
851:
816:
784:
740:
717:
688:
662:
626:
543:
519:
499:
469:
436:
416:
370:
337:
317:
293:
273:
235:
207:
181:
151:
127:
107:
1346:is a locally constructible subset, then the set of
1010:is a sequence of finitely presented quasi-coherent
2607:Topological definition of (local) constructibility
2270:"Theorem 29.22.3 (Chevalley's Theorem) (tag 054K)"
2101:
2079:
2059:
2037:
1989:
1963:
1937:
1902:
1880:
1860:
1838:
1799:
1773:
1747:
1712:
1690:
1670:
1648:
1629:{\displaystyle f_{s}\colon X_{s}\rightarrow Y_{s}}
1628:
1575:
1549:
1523:
1503:
1471:
1448:
1409:
1364:
1338:
1312:
1277:
1246:
1220:
1189:
1165:
1130:
1059:
1033:
1002:
948:
893:
837:
802:
770:
723:
703:
674:
644:
549:
529:
505:
482:
455:
422:
402:
349:
323:
299:
279:
259:
213:
193:
163:
133:
113:
1320:is an finitely presented morphism of schemes and
1173:is an finitely presented morphism of schemes and
956:is an finitely presented morphism of schemes and
901:, which is not a variety, but is constructible.
1456:is locally constructible. (Corollary (9.5.4))
8:
888:
870:
864:
852:
2339:, § 9 Propriétés constructibles, pp. 54-94.
39:and related fields. A key result known as
2611:https://stacks.math.columbia.edu/tag/054H
2605:https://stacks.math.columbia.edu/tag/04ZC
2094:
2092:
2072:
2052:
2050:
2008:
2002:
1976:
1950:
1918:
1895:
1893:
1873:
1853:
1851:
1818:
1812:
1786:
1760:
1728:
1705:
1703:
1683:
1663:
1661:
1641:
1620:
1607:
1594:
1588:
1562:
1536:
1516:
1484:
1464:
1428:
1422:
1383:
1377:
1351:
1325:
1293:
1269:
1263:
1262:
1259:
1233:
1212:
1206:
1205:
1202:
1181:
1180:
1178:
1146:
1119:
1113:
1112:
1102:
1096:
1095:
1082:
1076:
1075:
1072:
1046:
1025:
1019:
1018:
1015:
990:
989:
979:
978:
965:
964:
961:
929:
894:{\displaystyle \{x\neq 0\}\cup \{x=y=0\}}
850:
815:
783:
762:
757:
747:
742:
739:
716:
687:
661:
625:
542:
521:
520:
518:
498:
474:
468:
447:
435:
415:
388:
378:
369:
336:
316:
292:
272:
234:
206:
180:
150:
126:
106:
908:open sets in the definition) were used.
682:is a locally constructible subset, then
83:. (A set is locally closed if it is the
2147:
1197:is a finitely presented quasi-coherent
1938:{\displaystyle f\colon X\rightarrow S}
1748:{\displaystyle f\colon X\rightarrow S}
1504:{\displaystyle f\colon X\rightarrow Y}
1313:{\displaystyle f\colon X\rightarrow S}
1166:{\displaystyle f\colon X\rightarrow S}
949:{\displaystyle f\colon X\rightarrow S}
7:
2579:Constructible sets with applications
2516:Publications Mathématiques de l'IHÉS
2474:Publications Mathématiques de l'IHÉS
2432:Publications Mathématiques de l'IHÉS
566:locally noetherian topological space
2380:Constructible sets in real geometry
522:
1278:{\displaystyle {\mathcal {F}}_{s}}
1221:{\displaystyle {\mathcal {O}}_{X}}
1034:{\displaystyle {\mathcal {O}}_{X}}
16:For a Gödel constructive set, see
14:
2349:Grothendieck & Dieudonné 1966
2333:Grothendieck & Dieudonné 1966
2293:Grothendieck & Dieudonné 1971
2253:Grothendieck & Dieudonné 1964
2234:Grothendieck & Dieudonné 1971
2198:Grothendieck & Dieudonné 1961
2155:Grothendieck & Dieudonné 1961
711:is also locally constructible in
2583:PWN-Polish Scientific Publishers
2095:
2053:
1896:
1854:
1706:
1664:
758:
743:
403:{\displaystyle (U_{i})_{i\in I}}
1583:for which the induced morphism
1410:{\displaystyle f^{-1}(s)\cap Z}
530:{\displaystyle {\mathfrak {C}}}
2175:"Definition 5.15.1 (tag 005G)"
2032:
2029:
2023:
2017:
1929:
1833:
1827:
1739:
1613:
1495:
1443:
1437:
1398:
1392:
1304:
1190:{\displaystyle {\mathcal {F}}}
1157:
1108:
1091:
985:
975:
940:
832:
817:
797:
785:
753:
698:
692:
636:
385:
371:
254:
242:
175:for every compact open subset
1:
463:is a constructible subset of
229:union of subsets of the form
2102:{\displaystyle \mathbf {P} }
2087:is locally constructible if
2060:{\displaystyle \mathbf {P} }
2038:{\displaystyle f^{-1}(f(x))}
1903:{\displaystyle \mathbf {P} }
1888:is locally constructible if
1861:{\displaystyle \mathbf {P} }
1713:{\displaystyle \mathbf {P} }
1698:is locally constructible if
1671:{\displaystyle \mathbf {P} }
1531:-schemes. Consider the set
588:open subset of its closure.
513:are the smallest collection
31:are a class of subsets of a
2299:, Théorème (7.1.4), p. 329.
2259:, Théorème (1.8.4), p. 239.
456:{\displaystyle Z\cap U_{i}}
260:{\displaystyle U\cap (X-V)}
2648:
1964:{\displaystyle P\subset X}
1774:{\displaystyle P\subset S}
1550:{\displaystyle P\subset S}
1339:{\displaystyle Z\subset X}
1041:-modules, then the set of
675:{\displaystyle Z\subset X}
350:{\displaystyle Z\subset X}
194:{\displaystyle U\subset X}
63:in algebraic geometry and
15:
1839:{\displaystyle f^{-1}(s)}
1449:{\displaystyle f^{-1}(s)}
1228:-module, then the set of
2413:Linear algebraic groups.
2314:stacks.math.columbia.edu
2274:stacks.math.columbia.edu
2179:stacks.math.columbia.edu
912:Constructible properties
656:morphism of schemes and
645:{\displaystyle f:X\to Y}
580:In any (not necessarily
2545:Grothendieck, Alexandre
2503:Grothendieck, Alexandre
2461:Grothendieck, Alexandre
2419:Grothendieck, Alexandre
1417:is closed (or open) in
605:
164:{\displaystyle Z\cap U}
121:of a topological space
65:intersection cohomology
2131:Constructible topology
2103:
2081:
2061:
2039:
1991:
1990:{\displaystyle x\in X}
1965:
1939:
1904:
1882:
1862:
1840:
1801:
1800:{\displaystyle s\in S}
1775:
1749:
1714:
1692:
1672:
1650:
1630:
1577:
1576:{\displaystyle s\in S}
1551:
1525:
1505:
1473:
1450:
1411:
1366:
1365:{\displaystyle s\in S}
1340:
1314:
1279:
1248:
1247:{\displaystyle s\in S}
1222:
1191:
1167:
1132:
1061:
1060:{\displaystyle s\in S}
1035:
1004:
950:
895:
839:
838:{\displaystyle (x,xy)}
804:
772:
725:
705:
676:
646:
551:
531:
507:
484:
457:
424:
404:
351:
325:
301:
281:
261:
215:
195:
165:
135:
115:
18:Constructible universe
2368:Allouche, Jean Paul.
2104:
2082:
2062:
2040:
1992:
1966:
1940:
1905:
1883:
1863:
1841:
1802:
1776:
1750:
1715:
1693:
1673:
1651:
1631:
1578:
1552:
1526:
1506:
1474:
1451:
1412:
1367:
1341:
1315:
1280:
1249:
1223:
1192:
1168:
1133:
1062:
1036:
1005:
951:
918:quasicoherent sheaves
896:
840:
805:
803:{\displaystyle (x,y)}
773:
726:
706:
677:
647:
552:
532:
508:
485:
483:{\displaystyle U_{i}}
458:
425:
405:
359:locally constructible
352:
326:
302:
282:
262:
216:
196:
166:
136:
116:
61:constructible sheaves
2207:, Sect. (9.1), p. 12
2091:
2071:
2049:
2001:
1997:for which the fibre
1975:
1949:
1917:
1892:
1872:
1850:
1811:
1807:for which the fibre
1785:
1759:
1727:
1702:
1682:
1660:
1640:
1587:
1561:
1535:
1515:
1483:
1463:
1421:
1376:
1350:
1324:
1292:
1258:
1232:
1201:
1177:
1145:
1071:
1045:
1014:
960:
928:
849:
814:
782:
738:
715:
704:{\displaystyle f(Z)}
686:
660:
624:
618:Chevalley's theorem.
541:
517:
497:
467:
434:
414:
368:
335:
315:
291:
271:
233:
205:
179:
149:
125:
105:
2585:. pp. ix+269.
2216:Jinpeng An (2012).
2136:Constructible sheaf
1127:
1090:
606:Chevalley's theorem
575:algebraic varieties
81:locally closed sets
55:(or more generally
53:algebraic varieties
47:(more specifically
41:Chevalley's theorem
2632:Algebraic geometry
2529:10.1007/bf02684343
2487:10.1007/bf02684747
2445:10.1007/bf02684274
2386:. pp. x+270.
2099:
2077:
2057:
2035:
1987:
1961:
1935:
1900:
1878:
1858:
1836:
1797:
1771:
1745:
1710:
1688:
1668:
1656:has some property
1646:
1626:
1573:
1547:
1521:
1501:
1469:
1446:
1407:
1362:
1336:
1310:
1275:
1244:
1218:
1187:
1163:
1128:
1111:
1074:
1057:
1031:
1000:
946:
891:
845:has image the set
835:
800:
768:
721:
701:
672:
654:finitely presented
642:
547:
527:
503:
480:
453:
420:
400:
347:
321:
297:
277:
257:
211:
191:
161:
131:
111:
37:algebraic geometry
29:constructible sets
2566:978-3-540-05113-8
2220:. Geom. Dedicata
2080:{\displaystyle P}
1881:{\displaystyle P}
1691:{\displaystyle P}
1649:{\displaystyle s}
1524:{\displaystyle S}
1472:{\displaystyle S}
724:{\displaystyle Y}
550:{\displaystyle X}
506:{\displaystyle X}
423:{\displaystyle X}
324:{\displaystyle X}
300:{\displaystyle V}
280:{\displaystyle U}
214:{\displaystyle X}
134:{\displaystyle X}
114:{\displaystyle Z}
33:topological space
2639:
2594:
2570:
2540:
2498:
2456:
2405:
2356:
2346:
2340:
2330:
2324:
2323:
2321:
2320:
2306:
2300:
2290:
2284:
2283:
2281:
2280:
2266:
2260:
2250:
2244:
2231:
2225:
2214:
2208:
2195:
2189:
2188:
2186:
2185:
2171:
2165:
2152:
2108:
2106:
2105:
2100:
2098:
2086:
2084:
2083:
2078:
2066:
2064:
2063:
2058:
2056:
2044:
2042:
2041:
2036:
2016:
2015:
1996:
1994:
1993:
1988:
1970:
1968:
1967:
1962:
1944:
1942:
1941:
1936:
1909:
1907:
1906:
1901:
1899:
1887:
1885:
1884:
1879:
1867:
1865:
1864:
1859:
1857:
1845:
1843:
1842:
1837:
1826:
1825:
1806:
1804:
1803:
1798:
1780:
1778:
1777:
1772:
1754:
1752:
1751:
1746:
1719:
1717:
1716:
1711:
1709:
1697:
1695:
1694:
1689:
1677:
1675:
1674:
1669:
1667:
1655:
1653:
1652:
1647:
1635:
1633:
1632:
1627:
1625:
1624:
1612:
1611:
1599:
1598:
1582:
1580:
1579:
1574:
1556:
1554:
1553:
1548:
1530:
1528:
1527:
1522:
1510:
1508:
1507:
1502:
1479:be a scheme and
1478:
1476:
1475:
1470:
1455:
1453:
1452:
1447:
1436:
1435:
1416:
1414:
1413:
1408:
1391:
1390:
1371:
1369:
1368:
1363:
1345:
1343:
1342:
1337:
1319:
1317:
1316:
1311:
1284:
1282:
1281:
1276:
1274:
1273:
1268:
1267:
1253:
1251:
1250:
1245:
1227:
1225:
1224:
1219:
1217:
1216:
1211:
1210:
1196:
1194:
1193:
1188:
1186:
1185:
1172:
1170:
1169:
1164:
1137:
1135:
1134:
1129:
1123:
1118:
1117:
1107:
1106:
1101:
1100:
1086:
1081:
1080:
1066:
1064:
1063:
1058:
1040:
1038:
1037:
1032:
1030:
1029:
1024:
1023:
1009:
1007:
1006:
1001:
999:
995:
994:
984:
983:
974:
970:
969:
955:
953:
952:
947:
900:
898:
897:
892:
844:
842:
841:
836:
809:
807:
806:
801:
777:
775:
774:
769:
767:
766:
761:
752:
751:
746:
730:
728:
727:
722:
710:
708:
707:
702:
681:
679:
678:
673:
651:
649:
648:
643:
556:
554:
553:
548:
536:
534:
533:
528:
526:
525:
512:
510:
509:
504:
489:
487:
486:
481:
479:
478:
462:
460:
459:
454:
452:
451:
429:
427:
426:
421:
409:
407:
406:
401:
399:
398:
383:
382:
356:
354:
353:
348:
330:
328:
327:
322:
309:and retrocompact
306:
304:
303:
298:
286:
284:
283:
278:
266:
264:
263:
258:
220:
218:
217:
212:
200:
198:
197:
192:
170:
168:
167:
162:
140:
138:
137:
132:
120:
118:
117:
112:
2647:
2646:
2642:
2641:
2640:
2638:
2637:
2636:
2617:
2616:
2601:
2573:
2567:
2557:Springer-Verlag
2549:Dieudonné, Jean
2543:
2507:Dieudonné, Jean
2501:
2465:Dieudonné, Jean
2459:
2423:Dieudonné, Jean
2417:
2394:
2384:Springer-Verlag
2377:
2365:
2360:
2359:
2347:
2343:
2331:
2327:
2318:
2316:
2308:
2307:
2303:
2291:
2287:
2278:
2276:
2268:
2267:
2263:
2251:
2247:
2242:
2232:
2228:
2215:
2211:
2206:
2196:
2192:
2183:
2181:
2173:
2172:
2168:
2163:
2153:
2149:
2144:
2127:
2089:
2088:
2069:
2068:
2047:
2046:
2045:has a property
2004:
1999:
1998:
1973:
1972:
1947:
1946:
1915:
1914:
1890:
1889:
1870:
1869:
1848:
1847:
1846:has a property
1814:
1809:
1808:
1783:
1782:
1757:
1756:
1725:
1724:
1700:
1699:
1680:
1679:
1658:
1657:
1638:
1637:
1636:of fibres over
1616:
1603:
1590:
1585:
1584:
1559:
1558:
1533:
1532:
1513:
1512:
1481:
1480:
1461:
1460:
1424:
1419:
1418:
1379:
1374:
1373:
1348:
1347:
1322:
1321:
1290:
1289:
1261:
1256:
1255:
1230:
1229:
1204:
1199:
1198:
1175:
1174:
1143:
1142:
1094:
1069:
1068:
1043:
1042:
1017:
1012:
1011:
988:
963:
958:
957:
926:
925:
914:
847:
846:
812:
811:
780:
779:
756:
741:
736:
735:
713:
712:
684:
683:
658:
657:
622:
621:
608:
539:
538:
515:
514:
495:
494:
470:
465:
464:
443:
432:
431:
412:
411:
384:
374:
366:
365:
333:
332:
313:
312:
289:
288:
269:
268:
231:
230:
203:
202:
177:
176:
147:
146:
123:
122:
103:
102:
73:
21:
12:
11:
5:
2645:
2643:
2635:
2634:
2629:
2619:
2618:
2615:
2614:
2608:
2600:
2599:External links
2597:
2596:
2595:
2571:
2565:
2541:
2499:
2457:
2415:
2406:
2392:
2375:
2364:
2361:
2358:
2357:
2341:
2325:
2301:
2285:
2261:
2245:
2240:
2226:
2209:
2204:
2190:
2166:
2161:
2146:
2145:
2143:
2140:
2139:
2138:
2133:
2126:
2123:
2111:
2110:
2097:
2076:
2055:
2034:
2031:
2028:
2025:
2022:
2019:
2014:
2011:
2007:
1986:
1983:
1980:
1960:
1957:
1954:
1934:
1931:
1928:
1925:
1922:
1911:
1898:
1877:
1856:
1835:
1832:
1829:
1824:
1821:
1817:
1796:
1793:
1790:
1770:
1767:
1764:
1744:
1741:
1738:
1735:
1732:
1721:
1708:
1687:
1666:
1645:
1623:
1619:
1615:
1610:
1606:
1602:
1597:
1593:
1572:
1569:
1566:
1546:
1543:
1540:
1520:
1511:a morphism of
1500:
1497:
1494:
1491:
1488:
1468:
1457:
1445:
1442:
1439:
1434:
1431:
1427:
1406:
1403:
1400:
1397:
1394:
1389:
1386:
1382:
1361:
1358:
1355:
1335:
1332:
1329:
1309:
1306:
1303:
1300:
1297:
1286:
1272:
1266:
1243:
1240:
1237:
1215:
1209:
1184:
1162:
1159:
1156:
1153:
1150:
1139:
1126:
1122:
1116:
1110:
1105:
1099:
1093:
1089:
1085:
1079:
1056:
1053:
1050:
1028:
1022:
998:
993:
987:
982:
977:
973:
968:
945:
942:
939:
936:
933:
913:
910:
890:
887:
884:
881:
878:
875:
872:
869:
866:
863:
860:
857:
854:
834:
831:
828:
825:
822:
819:
799:
796:
793:
790:
787:
765:
760:
755:
750:
745:
720:
700:
697:
694:
691:
671:
668:
665:
641:
638:
635:
632:
629:
607:
604:
600:Stacks Project
546:
537:of subsets of
524:
502:
477:
473:
450:
446:
442:
439:
419:
397:
394:
391:
387:
381:
377:
373:
361:if there is a
346:
343:
340:
320:
296:
276:
256:
253:
250:
247:
244:
241:
238:
210:
201:. A subset of
190:
187:
184:
160:
157:
154:
130:
110:
72:
69:
13:
10:
9:
6:
4:
3:
2:
2644:
2633:
2630:
2628:
2625:
2624:
2622:
2612:
2609:
2606:
2603:
2602:
2598:
2592:
2588:
2584:
2580:
2576:
2575:Mostowski, A.
2572:
2568:
2562:
2558:
2554:
2550:
2546:
2542:
2538:
2534:
2530:
2526:
2522:
2518:
2517:
2512:
2508:
2504:
2500:
2496:
2492:
2488:
2484:
2480:
2476:
2475:
2470:
2466:
2462:
2458:
2454:
2450:
2446:
2442:
2438:
2434:
2433:
2428:
2424:
2420:
2416:
2414:
2410:
2409:Borel, Armand
2407:
2403:
2399:
2395:
2393:3-540-60451-0
2389:
2385:
2381:
2376:
2374:
2372:
2367:
2366:
2362:
2354:
2350:
2345:
2342:
2338:
2334:
2329:
2326:
2315:
2311:
2305:
2302:
2298:
2294:
2289:
2286:
2275:
2271:
2265:
2262:
2258:
2254:
2249:
2246:
2239:
2235:
2230:
2227:
2223:
2219:
2213:
2210:
2203:
2199:
2194:
2191:
2180:
2176:
2170:
2167:
2160:
2156:
2151:
2148:
2141:
2137:
2134:
2132:
2129:
2128:
2124:
2122:
2120:
2116:
2074:
2026:
2020:
2012:
2009:
2005:
1984:
1981:
1978:
1958:
1955:
1952:
1932:
1926:
1923:
1920:
1912:
1875:
1830:
1822:
1819:
1815:
1794:
1791:
1788:
1768:
1765:
1762:
1742:
1736:
1733:
1730:
1722:
1685:
1643:
1621:
1617:
1608:
1604:
1600:
1595:
1591:
1570:
1567:
1564:
1544:
1541:
1538:
1518:
1498:
1492:
1489:
1486:
1466:
1458:
1440:
1432:
1429:
1425:
1404:
1401:
1395:
1387:
1384:
1380:
1359:
1356:
1353:
1333:
1330:
1327:
1307:
1301:
1298:
1295:
1287:
1270:
1241:
1238:
1235:
1213:
1160:
1154:
1151:
1148:
1140:
1124:
1120:
1103:
1087:
1083:
1054:
1051:
1048:
1026:
996:
971:
943:
937:
934:
931:
923:
922:
921:
919:
911:
909:
907:
902:
885:
882:
879:
876:
873:
867:
861:
858:
855:
829:
826:
823:
820:
794:
791:
788:
763:
748:
732:
718:
695:
689:
669:
666:
663:
655:
639:
633:
630:
627:
619:
615:
613:
603:
601:
597:
593:
589:
587:
583:
578:
576:
571:
567:
562:
560:
544:
500:
491:
475:
471:
448:
444:
440:
437:
417:
395:
392:
389:
379:
375:
364:
360:
344:
341:
338:
318:
310:
294:
274:
251:
248:
245:
239:
236:
228:
224:
223:constructible
208:
188:
185:
182:
174:
158:
155:
152:
144:
128:
108:
100:
96:
94:
90:
86:
82:
78:
70:
68:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
19:
2578:
2552:
2520:
2514:
2478:
2472:
2436:
2430:
2412:
2379:
2369:
2352:
2344:
2336:
2328:
2317:. Retrieved
2313:
2304:
2296:
2288:
2277:. Retrieved
2273:
2264:
2256:
2248:
2237:
2229:
2221:
2212:
2201:
2193:
2182:. Retrieved
2178:
2169:
2158:
2150:
2118:
2112:
915:
906:retrocompact
905:
903:
733:
617:
616:
609:
592:Terminology:
591:
590:
579:
569:
563:
492:
358:
331:. A subset
308:
226:
222:
143:retrocompact
142:
99:Definitions:
98:
97:
85:intersection
74:
40:
28:
22:
778:that sends
559:complements
311:subsets of
267:where both
225:if it is a
71:Definitions
2621:Categories
2363:References
2319:2022-10-04
2279:2022-10-04
2224:: 153–185.
2184:2022-10-04
1372:for which
1254:for which
1067:for which
582:noetherian
141:is called
93:closed set
2010:−
1982:∈
1956:⊂
1930:→
1924::
1820:−
1792:∈
1766:⊂
1740:→
1734::
1614:→
1601::
1568:∈
1542:⊂
1496:→
1490::
1430:−
1402:∩
1385:−
1357:∈
1331:⊂
1305:→
1299::
1239:∈
1158:→
1152::
1109:→
1092:→
1052:∈
986:→
976:→
941:→
935::
868:∪
859:≠
754:→
667:⊂
637:→
441:∩
393:∈
342:⊂
307:are open
249:−
240:∩
186:⊂
156:∩
101:A subset
49:morphisms
2627:Topology
2577:(1969).
2551:(1971).
2509:(1966).
2467:(1964).
2425:(1961).
2125:See also
1125:″
1088:′
997:″
972:′
598:and the
89:open set
45:mappings
25:topology
2591:0255390
2537:0217086
2495:0173675
2453:0217085
2402:1393194
2067:. Then
1868:. Then
1678:. Then
173:compact
57:schemes
2589:
2563:
2535:
2493:
2451:
2400:
2390:
2351:, Ch.
2335:, Ch.
2295:, Ch.
2255:, Ch.
2236:, Ch.
2200:, Ch.
2157:, Ch.
227:finite
87:of an
2142:Notes
652:is a
612:image
586:dense
564:In a
363:cover
77:union
51:) of
2561:ISBN
2388:ISBN
2119:open
2115:flat
1913:Let
1723:Let
1459:Let
287:and
91:and
2525:doi
2483:doi
2441:doi
2222:157
2205:III
2162:III
1971:of
1781:of
1557:of
1288:If
1141:If
924:If
810:to
620:If
596:EGA
570:all
490:.
410:of
357:is
221:is
171:is
145:if
79:of
23:In
2623::
2587:MR
2559:.
2547:;
2533:MR
2531:.
2523:.
2521:28
2519:.
2513:.
2505:;
2491:MR
2489:.
2481:.
2479:20
2477:.
2471:.
2463:;
2449:MR
2447:.
2439:.
2437:11
2435:.
2429:.
2421:;
2411:.
2398:MR
2396:.
2353:IV
2337:IV
2312:.
2272:.
2177:.
731:.
568:,
67:.
27:,
2593:.
2569:.
2539:.
2527::
2497:.
2485::
2455:.
2443::
2404:.
2373:.
2322:.
2297:I
2282:.
2257:I
2241:I
2238:0
2202:0
2187:.
2159:0
2096:P
2075:P
2054:P
2033:)
2030:)
2027:x
2024:(
2021:f
2018:(
2013:1
2006:f
1985:X
1979:x
1959:X
1953:P
1933:S
1927:X
1921:f
1897:P
1876:P
1855:P
1834:)
1831:s
1828:(
1823:1
1816:f
1795:S
1789:s
1769:S
1763:P
1743:S
1737:X
1731:f
1707:P
1686:P
1665:P
1644:s
1622:s
1618:Y
1609:s
1605:X
1596:s
1592:f
1571:S
1565:s
1545:S
1539:P
1519:S
1499:Y
1493:X
1487:f
1467:S
1444:)
1441:s
1438:(
1433:1
1426:f
1405:Z
1399:)
1396:s
1393:(
1388:1
1381:f
1360:S
1354:s
1334:X
1328:Z
1308:S
1302:X
1296:f
1271:s
1265:F
1242:S
1236:s
1214:X
1208:O
1183:F
1161:S
1155:X
1149:f
1121:s
1115:F
1104:s
1098:F
1084:s
1078:F
1055:S
1049:s
1027:X
1021:O
992:F
981:F
967:F
944:S
938:X
932:f
889:}
886:0
883:=
880:y
877:=
874:x
871:{
865:}
862:0
856:x
853:{
833:)
830:y
827:x
824:,
821:x
818:(
798:)
795:y
792:,
789:x
786:(
764:2
759:A
749:2
744:A
719:Y
699:)
696:Z
693:(
690:f
670:X
664:Z
640:Y
634:X
631::
628:f
545:X
523:C
501:X
476:i
472:U
449:i
445:U
438:Z
418:X
396:I
390:i
386:)
380:i
376:U
372:(
345:X
339:Z
319:X
295:V
275:U
255:)
252:V
246:X
243:(
237:U
209:X
189:X
183:U
159:U
153:Z
129:X
109:Z
20:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.