785:
333:
1898:
664:
1466:
2094:
1733:
1104:
1630:
985:
858:
1683:. This breaks the intuition that finite dimensional schemes are necessarily Noetherian. Also, this example provides motivation for why studying schemes over a non-Noetherian base; that is, schemes
483:
1947:
1381:
1032:
68:
1815:
1288:
171:
141:
247:
Having a (locally) Noetherian hypothesis for a statement about schemes generally makes a lot of problems more accessible because they sufficiently rigidify many of its properties.
590:
442:
389:
219:
1957:
Another example of a non-Noetherian finite-dimensional scheme (in fact zero-dimensional) is given by the following quotient of a polynomial ring with infinitely many generators.
672:
1661:
1511:
1171:
413:
360:
1774:
901:
1213:
525:
96:
2362:
1582:
1823:
1681:
1534:
1273:
1253:
1233:
1124:
1052:
921:
603:
1389:
269:
338:
proving one of the sheaves has some property is equivalent to proving the other two have the property. In particular, given a fixed coherent sheaf
1963:
2111:
1686:
485:. Since this process can only be non-trivially applied only a finite number of times, this makes many induction arguments possible.
2266:
1057:
1587:
926:
796:
2332:
2294:
2126:
106:
if it is covered by spectra of
Noetherian rings. Thus, a scheme is Noetherian if and only if it is locally Noetherian and
2369:
2381:
1910:
1338:
990:
447:
1335:
In particular, infinitesimal deformations of
Noetherian schemes are again Noetherian. For example, given a curve
1291:. Also, this property can be used to show many schemes considered in algebraic geometry are in fact Noetherian.
40:
1468:
is also a
Noetherian scheme. A tower of such deformations can be used to construct formal Noetherian schemes.
2391:
1779:
780:{\displaystyle \varinjlim H^{i}(X,{\mathcal {F}}_{\alpha })\to H^{i}(X,\varinjlim {\mathcal {F}}_{\alpha })}
150:
120:
225:
561:
187:
2407:
2386:
1635:
1284:
1487:
2116:
418:
365:
1129:
394:
341:
2193:
1327:. Basically all of the objects from classical algebraic geometry fit into this class of examples.
255:
One of the most important structure theorems about
Noetherian rings and Noetherian schemes is the
1312:
228:. But the converse is false in general; consider, for example, the spectrum of a non-Noetherian
1545:
1514:
540:
528:
1741:
874:
1192:
504:
2342:
1893:{\displaystyle {\text{Spec}}({\mathcal {O}}_{K^{ur}})\to {\text{Spec}}({\mathcal {O}}_{K})}
74:
25:
2350:
8:
1559:
558:
agree on an affine open cover. This makes it possible to compute the sheaf cohomology of
2259:
659:{\displaystyle \{{\mathcal {F}}_{\alpha },\phi _{\alpha \beta }\}_{\alpha \in \Lambda }}
1904:
1666:
1519:
1384:
1258:
1238:
1218:
1109:
1037:
906:
37:
17:
1299:
In particular, quasi-projective varieties are
Noetherian schemes. This class includes
551:
2328:
2300:
2290:
2121:
1316:
1308:
666:
of sheaves of abelian groups on a
Noetherian scheme, there is a canonical isomorphism
2346:
2316:
1537:
555:
29:
2358:
1279:, i.e. with a fixed Hilbert polynomial. This is important because it implies many
2338:
2324:
1738:
One special case of such an extension is taking the maximal unramified extension
1461:{\displaystyle {\mathcal {C}}/{\text{Spec}}(\mathbb {F} _{q}/(\varepsilon ^{n}))}
1300:
99:
328:{\displaystyle 0\to {\mathcal {E}}'\to {\mathcal {E}}\to {\mathcal {E}}''\to 0,}
2105:
1482:
1304:
1276:
260:
229:
2401:
2304:
1324:
236:
107:
2234:
2210:
2169:
2145:
1541:
2089:{\displaystyle {\frac {\mathbb {Q} }{(x_{1},x_{2}^{2},x_{3}^{3},\ldots )}}}
1280:
111:
2284:
263:
into inductive arguments. Given a short exact sequence of coherent sheaves
256:
1189:
Another class of examples of
Noetherian schemes are families of schemes
2108:- slightly more rigid than Noetherian rings, but with better properties
1540:. There is a notion of algebraic geometry over such rings developed by
1320:
182:
496:
1536:. In order to deal with such rings, a topology is considered, giving
1275:. This includes many examples, such as the connected components of a
1330:
33:
1952:
2323:. Graduate Texts in Mathematics. Vol. 52. Berlin, New York:
1184:
493:
Every
Noetherian scheme can only have finitely many components.
117:
It can be shown that, in a locally
Noetherian scheme, if
1728:{\displaystyle {\text{Sch}}/{\text{Spec}}({\mathcal {O}}_{E})}
1551:
1099:{\displaystyle \mathbf {R} f_{*}({\mathcal {E}}^{\bullet })}
595:
1625:{\displaystyle \mathbb {Q} (\zeta _{\infty })/\mathbb {Q} }
980:{\displaystyle {\mathcal {E}}^{\bullet }\in D_{Coh}^{b}(X)}
1481:
One of the natural rings which are non-Noetherian are the
1632:(by adjoining all roots of unity), the ring of integers
853:{\displaystyle H^{i}(X,-):{\text{Ab}}(X)\to {\text{Ab}}}
110:. As with Noetherian rings, the concept is named after
987:
with bounded coherent cohomology such that the sheaves
242:
259:. This makes it possible to decompose arguments about
177:
is a
Noetherian ring. For a locally Noetherian scheme
1966:
1913:
1826:
1782:
1744:
1689:
1669:
1638:
1590:
1562:
1522:
1490:
1392:
1341:
1261:
1241:
1221:
1195:
1132:
1112:
1060:
1040:
993:
929:
909:
877:
799:
675:
606:
564:
507:
450:
421:
397:
368:
344:
272:
190:
153:
123:
77:
43:
1283:
encountered in the wild are Noetherian, such as the
2260:"Weil and Grothendieck Approaches to Adelic Points"
592:using ÄŚech cohomology for the standard open cover.
497:
Morphisms from Noetherian schemes are quasi-compact
488:
2289:. Berlin, Heidelberg: Springer Berlin Heidelberg.
2088:
1941:
1892:
1809:
1768:
1727:
1675:
1655:
1624:
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1528:
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1227:
1207:
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979:
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779:
658:
584:
519:
477:
436:
407:
383:
354:
327:
213:
165:
135:
90:
62:
1942:{\displaystyle {\text{Spec}}({\mathcal {O}}_{K})}
1376:{\displaystyle C/{\text{Spec}}(\mathbb {F} _{q})}
1181:Most schemes of interest are Noetherian schemes.
2399:
1331:Infinitesimal deformations of Noetherian schemes
1027:{\displaystyle H^{i}({\mathcal {E}}^{\bullet })}
1953:Polynomial ring with infinitely many generators
415:has some property can be reduced to looking at
1735:, can be an interesting and fruitful subject.
478:{\displaystyle {\mathcal {F}}/{\mathcal {F}}'}
1294:
1185:Locally of finite type over a Noetherian base
1663:is a Non-noetherian ring which is dimension
1476:
641:
607:
546:
2315:
1552:Rings of integers over infinite extensions
63:{\displaystyle \operatorname {Spec} A_{i}}
2235:"Lemma 29.15.6 (01T6)—The Stacks project"
2211:"Lemma 36.10.3 (08E2)—The Stacks project"
2112:Chevalley's theorem on constructible sets
1971:
1618:
1592:
1556:Given an infinite Galois field extension
1493:
1415:
1360:
596:Compatibility of colimits with cohomology
567:
534:
2282:
2170:"Lemma 28.5.8 (01P0)—The Stacks project"
2146:"Lemma 28.5.7 (0BA8)—The Stacks project"
501:Every morphism from a Noetherian scheme
2379:
1810:{\displaystyle {\mathcal {O}}_{K^{ur}}}
866:
863:preserve direct limits and coproducts.
2400:
173:is a Noetherian scheme if and only if
166:{\displaystyle \operatorname {Spec} A}
136:{\displaystyle \operatorname {Spec} A}
1776:and considering the ring of integers
1106:has bounded coherent cohomology over
871:Given a locally finite type morphism
147:is a Noetherian ring; in particular,
585:{\displaystyle \mathbb {P} _{S}^{n}}
243:Properties and Noetherian hypotheses
214:{\displaystyle {\mathcal {O}}_{X,x}}
13:
2357:
2272:from the original on 21 July 2018.
2257:
1925:
1876:
1838:
1786:
1711:
1656:{\displaystyle {\mathcal {O}}_{K}}
1642:
1604:
1395:
1082:
1010:
933:
763:
711:
651:
613:
543:properties of Noetherian schemes.
466:
453:
425:
400:
372:
347:
307:
296:
282:
194:
14:
2419:
2363:"Cohomology of Arithmetic Groups"
2283:Neukirch, JĂĽrgen (1999). "1.13".
2127:Nagata's compactification theorem
1506:{\displaystyle \mathbb {A} _{K}}
1062:
489:Number of irreducible components
1471:
1289:Moduli of stable vector bundles
1054:, then the derived pushforward
437:{\displaystyle {\mathcal {F}}'}
384:{\displaystyle {\mathcal {F}}'}
143:is an open affine subset, then
2276:
2251:
2227:
2203:
2186:
2162:
2138:
2080:
2025:
2020:
1975:
1936:
1919:
1887:
1870:
1862:
1859:
1832:
1722:
1705:
1609:
1596:
1455:
1452:
1439:
1431:
1425:
1410:
1370:
1355:
1199:
1166:{\displaystyle D_{Coh}^{b}(S)}
1160:
1154:
1093:
1076:
1021:
1004:
974:
968:
887:
842:
839:
833:
822:
810:
774:
738:
725:
722:
699:
511:
408:{\displaystyle {\mathcal {F}}}
355:{\displaystyle {\mathcal {F}}}
316:
301:
291:
276:
102:. More generally, a scheme is
1:
2132:
1126:, meaning it is an object in
250:
226:Noetherian topological space
7:
2387:Encyclopedia of Mathematics
2099:
1176:
221:are also Noetherian rings.
10:
2424:
1295:Quasi-projective varieties
1285:Moduli of algebraic curves
235:The definitions extend to
1477:Schemes over Adelic bases
1034:have proper support over
923:and a complex of sheaves
547:ÄŚech and sheaf cohomology
362:and a sub-coherent sheaf
224:A Noetherian scheme is a
2239:stacks.math.columbia.edu
2215:stacks.math.columbia.edu
2174:stacks.math.columbia.edu
2150:stacks.math.columbia.edu
1769:{\displaystyle K^{ur}/K}
896:{\displaystyle f:X\to S}
2380:Danilov, V.I. (2001) ,
2286:Algebraic Number Theory
2194:"Cohomology of Sheaves"
1255:is of finite type over
903:to a Noetherian scheme
2117:Zariski's main theorem
2097:
2090:
1943:
1901:
1894:
1817:. The induced morphism
1811:
1770:
1729:
1677:
1657:
1626:
1578:
1546:Alexander Grothendieck
1530:
1515:algebraic number field
1507:
1462:
1377:
1269:
1249:
1229:
1209:
1208:{\displaystyle X\to S}
1167:
1120:
1100:
1048:
1028:
981:
917:
897:
861:
854:
788:
781:
660:
600:Given a direct system
586:
535:Homological properties
521:
520:{\displaystyle X\to S}
479:
438:
409:
385:
356:
336:
329:
215:
167:
137:
92:
64:
2091:
1959:
1944:
1895:
1819:
1812:
1771:
1730:
1678:
1658:
1627:
1579:
1531:
1508:
1463:
1378:
1270:
1250:
1230:
1210:
1168:
1121:
1101:
1049:
1029:
982:
918:
898:
855:
792:
782:
668:
661:
587:
522:
480:
439:
410:
386:
357:
330:
265:
216:
168:
138:
93:
91:{\displaystyle A_{i}}
65:
28:that admits a finite
1964:
1911:
1824:
1780:
1742:
1687:
1667:
1636:
1588:
1560:
1520:
1488:
1390:
1339:
1259:
1239:
1219:
1193:
1130:
1110:
1058:
1038:
991:
927:
907:
875:
867:Derived direct image
797:
790:meaning the functors
673:
604:
562:
539:There are many nice
505:
448:
419:
395:
366:
342:
270:
188:
151:
121:
75:
41:
2382:"Noetherian scheme"
2073:
2055:
1577:{\displaystyle K/L}
1153:
967:
581:
2408:Algebraic geometry
2321:Algebraic Geometry
2086:
2059:
2041:
1939:
1905:universal covering
1890:
1807:
1766:
1725:
1673:
1653:
1622:
1574:
1526:
1503:
1458:
1373:
1313:calabi-yau schemes
1265:
1245:
1235:is Noetherian and
1225:
1205:
1163:
1133:
1116:
1096:
1044:
1024:
977:
947:
913:
893:
850:
777:
755:
684:
656:
582:
565:
517:
475:
434:
405:
381:
352:
325:
211:
163:
133:
104:locally Noetherian
88:
60:
18:algebraic geometry
2334:978-0-387-90244-9
2317:Hartshorne, Robin
2296:978-3-662-03983-0
2122:Dualizing complex
2084:
1917:
1868:
1830:
1703:
1693:
1676:{\displaystyle 1}
1538:topological rings
1529:{\displaystyle K}
1408:
1353:
1317:shimura varieties
1309:abelian varieties
1268:{\displaystyle S}
1248:{\displaystyle X}
1228:{\displaystyle S}
1119:{\displaystyle S}
1047:{\displaystyle S}
916:{\displaystyle S}
848:
831:
748:
677:
257:dévissage theorem
22:Noetherian scheme
2415:
2394:
2376:
2374:
2368:. Archived from
2367:
2354:
2309:
2308:
2280:
2274:
2273:
2271:
2264:
2255:
2249:
2248:
2246:
2245:
2231:
2225:
2224:
2222:
2221:
2207:
2201:
2200:
2198:
2190:
2184:
2183:
2181:
2180:
2166:
2160:
2159:
2157:
2156:
2142:
2095:
2093:
2092:
2087:
2085:
2083:
2072:
2067:
2054:
2049:
2037:
2036:
2023:
2013:
2012:
2000:
1999:
1987:
1986:
1974:
1968:
1948:
1946:
1945:
1940:
1935:
1934:
1929:
1928:
1918:
1915:
1899:
1897:
1896:
1891:
1886:
1885:
1880:
1879:
1869:
1866:
1858:
1857:
1856:
1855:
1842:
1841:
1831:
1828:
1816:
1814:
1813:
1808:
1806:
1805:
1804:
1803:
1790:
1789:
1775:
1773:
1772:
1767:
1762:
1757:
1756:
1734:
1732:
1731:
1726:
1721:
1720:
1715:
1714:
1704:
1701:
1699:
1694:
1691:
1682:
1680:
1679:
1674:
1662:
1660:
1659:
1654:
1652:
1651:
1646:
1645:
1631:
1629:
1628:
1623:
1621:
1616:
1608:
1607:
1595:
1583:
1581:
1580:
1575:
1570:
1535:
1533:
1532:
1527:
1512:
1510:
1509:
1504:
1502:
1501:
1496:
1467:
1465:
1464:
1459:
1451:
1450:
1438:
1424:
1423:
1418:
1409:
1406:
1404:
1399:
1398:
1382:
1380:
1379:
1374:
1369:
1368:
1363:
1354:
1351:
1349:
1301:algebraic curves
1274:
1272:
1271:
1266:
1254:
1252:
1251:
1246:
1234:
1232:
1231:
1226:
1214:
1212:
1211:
1206:
1172:
1170:
1169:
1164:
1152:
1147:
1125:
1123:
1122:
1117:
1105:
1103:
1102:
1097:
1092:
1091:
1086:
1085:
1075:
1074:
1065:
1053:
1051:
1050:
1045:
1033:
1031:
1030:
1025:
1020:
1019:
1014:
1013:
1003:
1002:
986:
984:
983:
978:
966:
961:
943:
942:
937:
936:
922:
920:
919:
914:
902:
900:
899:
894:
859:
857:
856:
851:
849:
846:
832:
829:
809:
808:
786:
784:
783:
778:
773:
772:
767:
766:
756:
737:
736:
721:
720:
715:
714:
698:
697:
685:
665:
663:
662:
657:
655:
654:
639:
638:
623:
622:
617:
616:
591:
589:
588:
583:
580:
575:
570:
556:sheaf cohomology
526:
524:
523:
518:
484:
482:
481:
476:
474:
470:
469:
462:
457:
456:
443:
441:
440:
435:
433:
429:
428:
414:
412:
411:
406:
404:
403:
390:
388:
387:
382:
380:
376:
375:
361:
359:
358:
353:
351:
350:
334:
332:
331:
326:
315:
311:
310:
300:
299:
290:
286:
285:
261:coherent sheaves
220:
218:
217:
212:
210:
209:
198:
197:
172:
170:
169:
164:
142:
140:
139:
134:
97:
95:
94:
89:
87:
86:
69:
67:
66:
61:
59:
58:
2423:
2422:
2418:
2417:
2416:
2414:
2413:
2412:
2398:
2397:
2372:
2365:
2335:
2325:Springer-Verlag
2312:
2297:
2281:
2277:
2269:
2262:
2258:Conrad, Brian.
2256:
2252:
2243:
2241:
2233:
2232:
2228:
2219:
2217:
2209:
2208:
2204:
2196:
2192:
2191:
2187:
2178:
2176:
2168:
2167:
2163:
2154:
2152:
2144:
2143:
2139:
2135:
2102:
2068:
2063:
2050:
2045:
2032:
2028:
2024:
2008:
2004:
1995:
1991:
1982:
1978:
1970:
1969:
1967:
1965:
1962:
1961:
1955:
1930:
1924:
1923:
1922:
1914:
1912:
1909:
1908:
1881:
1875:
1874:
1873:
1865:
1848:
1844:
1843:
1837:
1836:
1835:
1827:
1825:
1822:
1821:
1796:
1792:
1791:
1785:
1784:
1783:
1781:
1778:
1777:
1758:
1749:
1745:
1743:
1740:
1739:
1716:
1710:
1709:
1708:
1700:
1695:
1690:
1688:
1685:
1684:
1668:
1665:
1664:
1647:
1641:
1640:
1639:
1637:
1634:
1633:
1617:
1612:
1603:
1599:
1591:
1589:
1586:
1585:
1566:
1561:
1558:
1557:
1554:
1521:
1518:
1517:
1497:
1492:
1491:
1489:
1486:
1485:
1479:
1474:
1446:
1442:
1434:
1419:
1414:
1413:
1405:
1400:
1394:
1393:
1391:
1388:
1387:
1364:
1359:
1358:
1350:
1345:
1340:
1337:
1336:
1333:
1305:elliptic curves
1297:
1260:
1257:
1256:
1240:
1237:
1236:
1220:
1217:
1216:
1215:where the base
1194:
1191:
1190:
1187:
1179:
1148:
1137:
1131:
1128:
1127:
1111:
1108:
1107:
1087:
1081:
1080:
1079:
1070:
1066:
1061:
1059:
1056:
1055:
1039:
1036:
1035:
1015:
1009:
1008:
1007:
998:
994:
992:
989:
988:
962:
951:
938:
932:
931:
930:
928:
925:
924:
908:
905:
904:
876:
873:
872:
869:
845:
828:
804:
800:
798:
795:
794:
768:
762:
761:
760:
747:
732:
728:
716:
710:
709:
708:
693:
689:
676:
674:
671:
670:
644:
640:
631:
627:
618:
612:
611:
610:
605:
602:
601:
598:
576:
571:
566:
563:
560:
559:
552:ÄŚech cohomology
549:
537:
506:
503:
502:
499:
491:
465:
464:
463:
458:
452:
451:
449:
446:
445:
424:
423:
422:
420:
417:
416:
399:
398:
396:
393:
392:
371:
370:
369:
367:
364:
363:
346:
345:
343:
340:
339:
306:
305:
304:
295:
294:
281:
280:
279:
271:
268:
267:
253:
245:
199:
193:
192:
191:
189:
186:
185:
152:
149:
148:
122:
119:
118:
100:Noetherian ring
82:
78:
76:
73:
72:
54:
50:
42:
39:
38:
36:affine subsets
12:
11:
5:
2421:
2411:
2410:
2396:
2395:
2377:
2375:on 2020-07-24.
2359:Harder, GĂĽnter
2355:
2333:
2311:
2310:
2295:
2275:
2250:
2226:
2202:
2185:
2161:
2136:
2134:
2131:
2130:
2129:
2124:
2119:
2114:
2109:
2106:Excellent ring
2101:
2098:
2082:
2079:
2076:
2071:
2066:
2062:
2058:
2053:
2048:
2044:
2040:
2035:
2031:
2027:
2022:
2019:
2016:
2011:
2007:
2003:
1998:
1994:
1990:
1985:
1981:
1977:
1973:
1954:
1951:
1938:
1933:
1927:
1921:
1889:
1884:
1878:
1872:
1864:
1861:
1854:
1851:
1847:
1840:
1834:
1802:
1799:
1795:
1788:
1765:
1761:
1755:
1752:
1748:
1724:
1719:
1713:
1707:
1698:
1672:
1650:
1644:
1620:
1615:
1611:
1606:
1602:
1598:
1594:
1573:
1569:
1565:
1553:
1550:
1525:
1500:
1495:
1483:Ring of adeles
1478:
1475:
1473:
1470:
1457:
1454:
1449:
1445:
1441:
1437:
1433:
1430:
1427:
1422:
1417:
1412:
1403:
1397:
1372:
1367:
1362:
1357:
1348:
1344:
1332:
1329:
1325:cubic surfaces
1296:
1293:
1277:Hilbert scheme
1264:
1244:
1224:
1204:
1201:
1198:
1186:
1183:
1178:
1175:
1162:
1159:
1156:
1151:
1146:
1143:
1140:
1136:
1115:
1095:
1090:
1084:
1078:
1073:
1069:
1064:
1043:
1023:
1018:
1012:
1006:
1001:
997:
976:
973:
970:
965:
960:
957:
954:
950:
946:
941:
935:
912:
892:
889:
886:
883:
880:
868:
865:
844:
841:
838:
835:
827:
824:
821:
818:
815:
812:
807:
803:
776:
771:
765:
759:
754:
751:
746:
743:
740:
735:
731:
727:
724:
719:
713:
707:
704:
701:
696:
692:
688:
683:
680:
653:
650:
647:
643:
637:
634:
630:
626:
621:
615:
609:
597:
594:
579:
574:
569:
548:
545:
536:
533:
516:
513:
510:
498:
495:
490:
487:
473:
468:
461:
455:
432:
427:
402:
379:
374:
349:
324:
321:
318:
314:
309:
303:
298:
293:
289:
284:
278:
275:
252:
249:
244:
241:
237:formal schemes
230:valuation ring
208:
205:
202:
196:
162:
159:
156:
132:
129:
126:
85:
81:
57:
53:
49:
46:
9:
6:
4:
3:
2:
2420:
2409:
2406:
2405:
2403:
2393:
2389:
2388:
2383:
2378:
2371:
2364:
2360:
2356:
2352:
2348:
2344:
2340:
2336:
2330:
2326:
2322:
2318:
2314:
2313:
2306:
2302:
2298:
2292:
2288:
2287:
2279:
2268:
2261:
2254:
2240:
2236:
2230:
2216:
2212:
2206:
2195:
2189:
2175:
2171:
2165:
2151:
2147:
2141:
2137:
2128:
2125:
2123:
2120:
2118:
2115:
2113:
2110:
2107:
2104:
2103:
2096:
2077:
2074:
2069:
2064:
2060:
2056:
2051:
2046:
2042:
2038:
2033:
2029:
2017:
2014:
2009:
2005:
2001:
1996:
1992:
1988:
1983:
1979:
1958:
1950:
1931:
1906:
1900:
1882:
1852:
1849:
1845:
1818:
1800:
1797:
1793:
1763:
1759:
1753:
1750:
1746:
1736:
1717:
1696:
1670:
1648:
1613:
1600:
1571:
1567:
1563:
1549:
1547:
1543:
1539:
1523:
1516:
1498:
1484:
1469:
1447:
1443:
1435:
1428:
1420:
1401:
1386:
1365:
1346:
1342:
1328:
1326:
1322:
1318:
1314:
1310:
1306:
1302:
1292:
1290:
1286:
1282:
1281:moduli spaces
1278:
1262:
1242:
1222:
1202:
1196:
1182:
1174:
1157:
1149:
1144:
1141:
1138:
1134:
1113:
1088:
1071:
1067:
1041:
1016:
999:
995:
971:
963:
958:
955:
952:
948:
944:
939:
910:
890:
884:
881:
878:
864:
860:
836:
825:
819:
816:
813:
805:
801:
791:
787:
769:
757:
752:
749:
744:
741:
733:
729:
717:
705:
702:
694:
690:
686:
681:
678:
667:
648:
645:
635:
632:
628:
624:
619:
593:
577:
572:
557:
553:
544:
542:
532:
530:
529:quasi-compact
514:
508:
494:
486:
471:
459:
430:
377:
335:
322:
319:
312:
287:
273:
264:
262:
258:
248:
240:
238:
233:
231:
227:
222:
206:
203:
200:
184:
180:
176:
160:
157:
154:
146:
130:
127:
124:
115:
113:
109:
105:
101:
83:
79:
71:, where each
70:
55:
51:
47:
44:
35:
31:
27:
23:
19:
2385:
2370:the original
2320:
2285:
2278:
2253:
2242:. Retrieved
2238:
2229:
2218:. Retrieved
2214:
2205:
2188:
2177:. Retrieved
2173:
2164:
2153:. Retrieved
2149:
2140:
1960:
1956:
1902:
1820:
1737:
1555:
1480:
1472:Non-examples
1334:
1298:
1188:
1180:
870:
862:
793:
789:
669:
599:
550:
538:
500:
492:
337:
266:
254:
246:
234:
223:
178:
174:
144:
116:
112:Emmy Noether
103:
21:
15:
1385:deformation
1321:K3 surfaces
541:homological
183:local rings
2351:0367.14001
2244:2020-07-24
2220:2020-07-24
2179:2020-07-24
2155:2020-07-24
2133:References
1903:forms the
1584:, such as
391:, showing
2392:EMS Press
2305:851391469
2078:…
2018:…
1863:→
1605:∞
1601:ζ
1444:ε
1429:ε
1200:→
1089:∙
1072:∗
1017:∙
945:∈
940:∙
888:→
843:→
820:−
770:α
758:
753:→
726:→
718:α
687:
682:→
652:Λ
649:∈
646:α
636:β
633:α
629:ϕ
620:α
512:→
317:→
302:→
292:→
277:→
251:DĂ©vissage
158:
128:
48:
2402:Category
2319:(1977).
2267:Archived
2100:See also
1177:Examples
472:′
431:′
378:′
313:″
288:′
30:covering
2343:0463157
1513:for an
108:compact
2349:
2341:
2331:
2303:
2293:
1383:, any
1323:, and
26:scheme
2373:(PDF)
2366:(PDF)
2270:(PDF)
2263:(PDF)
2197:(PDF)
98:is a
24:is a
2329:ISBN
2301:OCLC
2291:ISBN
1916:Spec
1867:Spec
1829:Spec
1702:Spec
1544:and
1542:Weil
1407:Spec
1352:Spec
1287:and
554:and
444:and
181:the
155:Spec
125:Spec
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34:open
20:, a
2347:Zbl
1907:of
1692:Sch
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679:lim
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2390:,
2384:,
2361:.
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2337:.
2327:.
2299:.
2265:.
2237:.
2213:.
2172:.
2148:.
1949:.
1548:.
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1315:,
1311:,
1307:,
1303:,
1173:.
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830:Ab
531:.
239:.
232:.
179:X,
114:.
2353:.
2307:.
2247:.
2223:.
2199:.
2182:.
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2075:,
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2065:3
2061:x
2057:,
2052:2
2047:2
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2039:,
2034:1
2030:x
2026:(
2021:]
2015:,
2010:3
2006:x
2002:,
1997:2
1993:x
1989:,
1984:1
1980:x
1976:[
1972:Q
1937:)
1932:K
1926:O
1920:(
1888:)
1883:K
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1871:(
1860:)
1853:r
1850:u
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1440:(
1436:/
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