33:
2092:
is regular. (Zariski observed that this can fail over non-perfect fields.) This implies that smoothness is an intrinsic property of the variety, in other words it does not depend on where or how the variety is embedded in affine space. It also suggests that regular local rings should have good
1472:
2125:-module has a projective resolution of finite length. It is easy to show that the property of having finite global dimension is preserved under localization, and consequently that localizations of regular local rings at prime ideals are again regular.
741:
660:
2104:
Another property suggested by geometric intuition is that the localization of a regular local ring should again be regular. Again, this lay unsolved until the introduction of homological techniques. It was
976:
1555:
368:
1649:
1343:
990:
is regular. (The converse is always true: the multiplicity of a regular local ring is one.) This criterion corresponds to a geometric intuition in algebraic geometry that a local ring of an
849:
2387:
Herrmann, M., S. Ikeda, and U. Orbanz: Equimultiplicity and
Blowing Up. An Algebraic Study with an Appendix by B. Moonen. Springer Verlag, Berlin Heidelberg New-York, 1988. Theorem 6.8.
893:
1075:
194:
1917:
1805:
1606:
545:
498:
1739:
302:
679:
1335:
447:
1875:
1953:
2297:
1709:
1689:
1669:
1575:
1495:
1099:
1023:
809:
789:
765:
585:
565:
408:
388:
278:
2097:
very little was known in this direction. Once such techniques were introduced in the 1950s, Auslander and
Buchsbaum proved that every regular local ring is a
2167:
is a regular local ring: that is, every such localization has the property that the minimal number of generators of its maximal ideal is equal to its
1337:
is not a regular local ring since it is finite dimensional but does not have finite global dimension. For example, there is an infinite resolution
1125:
is a regular local ring of dimension 1 and the regular local rings of dimension 1 are exactly the discrete valuation rings. Specifically, if
2211:
global homological dimension. His definition is stronger than the definition above, which allows regular rings of infinite Krull dimension.
596:
2313:
but need not be an integral domain. For example, the product of two regular integral domains is regular, but not an integral domain.
2060:) is the matrix of partial derivatives of the defining equations of the variety, then the rank of the matrix found by evaluating
1118:
is a regular local ring. These have (Krull) dimension 0. In fact, the fields are exactly the regular local rings of dimension 0.
917:
152:
2554:
2187:
1500:
311:
1963:
1611:
1467:{\displaystyle \cdots {\xrightarrow {\cdot x}}{\frac {k}{(x^{2})}}{\xrightarrow {\cdot x}}{\frac {k}{(x^{2})}}\to k\to 0}
1230:
114:
2576:
260:
There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if
820:
1105:. No similar result is known in characteristic zero (it is unclear how one should replace the Frobenius morphism).
1151:
is an example of a discrete valuation ring, and consequently a regular local ring, which does not contain a field.
1959:
1750:
979:
910:
is unimixed (in the sense that there is no embedded prime divisor of the zero ideal and for each minimal prime
859:
991:
2198:
1988:
a few years later, who showed that geometrically, a regular local ring corresponds to a smooth point on an
1035:
995:
17:
162:
2597:
2300:
2098:
1884:
1774:
1754:
1580:
512:
457:
2322:
1714:
242:
82:
42:
2535:
Kunz, Characterizations of regular local rings of characteristic p. Amer. J. Math. 91 (1969), 772â784.
736:{\displaystyle {\mbox{gl dim }}A:=\sup\{\operatorname {pd} M\mid M{\text{ is an }}A{\text{-module}}\}}
2191:
283:
224:
2346:
A local von
Neumann regular ring is a division ring, so the two conditions are not very compatible.
2141:
1765:
1249:
1002:
213:
1284:
1241:
1122:
230:
51:
419:
2238:
2160:
1813:
1761:
1078:
1926:
2602:
2529:
903:
768:
8:
2183:
2148:
2094:
1197:
1134:
1115:
67:
2446:
2247:
1694:
1674:
1654:
1560:
1480:
1237:) ) is an example of a 2-dimensional regular local ring which does not contain a field.
1084:
1030:
1008:
794:
774:
750:
570:
550:
393:
373:
263:
197:
2480:
2572:
2560:
2550:
2204:
2106:
1997:
1989:
145:
47:
2513:
2475:
2438:
2409:
2118:
2039:
1005:
case, there is the following important result due to Kunz: A Noetherian local ring
744:
2568:
2546:
2525:
2327:
2242:
2215:
2168:
2156:
1265:
668:
236:
90:
75:
2521:
2509:
2397:
2179:
1981:
1148:
205:
2109:
who found a homological characterization of regular local rings: A local ring
2591:
2538:
2463:
2426:
2355:
since a ring is reduced if and only if its localizations at prime ideals are.
2005:
1985:
1975:
1920:
454:
where the dimension is the Krull dimension. The minimal set of generators of
156:
86:
2582:
2310:
1102:
2466:(1947), "The concept of a simple point of an abstract algebraic variety",
2369:
2493:
2164:
1245:
2400:(1937), "BeitrÀge zur Arithmetik kommutativer IntegritÀtsbereiche III",
219:
For
Noetherian local rings, there is the following chain of inclusions:
2450:
2413:
78:
2429:(1940), "Algebraic varieties over ground fields of characteristic 0",
97:
be any
Noetherian local ring with unique maximal ideal m, and suppose
655:{\displaystyle \dim _{k}{\mathfrak {m}}/{\mathfrak {m}}^{2}=\dim A\,}
2442:
2140:
For the unrelated regular rings introduced by John von
Neumann, see
1406:
1355:
2214:
Examples of regular rings include fields (of dimension zero) and
1807:
is a complete regular local ring that contains a field, then
2207:
defined a regular ring as a commutative noetherian ring of
2132:
for non-local commutative rings given in the next section.
2093:
properties, but before the introduction of techniques from
140:
The concept is motivated by its geometric meaning. A point
1181:
are indeterminates, then the ring of formal power series
971:{\displaystyle \dim {\widehat {A}}/p=\dim {\widehat {A}}}
1984:
in 1937, but they first became prominent in the work of
2306:
Any localization of a regular ring is regular as well.
1550:{\displaystyle {\mathfrak {m}}={\frac {(x)}{(x^{2})}}}
864:
825:
684:
2250:
1929:
1887:
1816:
1777:
1717:
1697:
1677:
1657:
1614:
1583:
1563:
1503:
1483:
1346:
1287:
1140:] is a regular local ring having (Krull) dimension 1.
1087:
1038:
1011:
920:
862:
823:
797:
777:
753:
682:
599:
573:
553:
515:
460:
422:
396:
376:
363:{\displaystyle {\mathfrak {m}}=(a_{1},\ldots ,a_{n})}
314:
286:
266:
165:
1644:{\displaystyle {\mathfrak {m}}/{\mathfrak {m}}^{2}}
1185:] is a regular local ring having (Krull) dimension
2291:
1947:
1911:
1869:
1799:
1733:
1703:
1683:
1663:
1643:
1600:
1569:
1549:
1489:
1466:
1329:
1196:is a regular local ring, then it follows that the
1093:
1069:
1017:
970:
887:
843:
803:
783:
759:
735:
654:
579:
559:
539:
492:
441:
402:
382:
362:
296:
272:
188:
304:, then the following are equivalent definitions:
2589:
994:is regular if and only if the intersection is a
696:
2508:
2375:
2197:For regular rings, Krull dimension agrees with
1980:Regular local rings were originally defined by
81:having the property that the minimal number of
844:{\displaystyle {\mbox{gl dim }}A<\infty \,}
280:is a Noetherian local ring with maximal ideal
2299:is regular. In the case of a field, this is
1477:Using another one of the characterizations,
730:
699:
2226:, with dimension one greater than that of
2012:is the vanishing locus of the polynomials
1969:
1753:states that every regular local ring is a
113:is a minimal set of generators of m. Then
2479:
1147:is an ordinary prime number, the ring of
840:
651:
438:
888:{\displaystyle {\mbox{gl dim }}A=\dim A}
2462:
2425:
2237:is a field, the ring of integers, or a
14:
2590:
1768:, of a regular local ring is regular.
1252:regular local ring of Krull dimension
1133:is an indeterminate, then the ring of
2396:
1070:{\displaystyle R\to R,r\mapsto r^{p}}
390:is chosen as small as possible. Then
255:
189:{\displaystyle {\mathcal {O}}_{X,x}}
26:
2583:Regular rings at The Stacks Project
2518:Introduction to Commutative Algebra
1912:{\displaystyle k=A/{\mathfrak {m}}}
1904:
1800:{\displaystyle (A,{\mathfrak {m}})}
1789:
1744:
1726:
1630:
1617:
1601:{\displaystyle {\mathfrak {m}}^{2}}
1587:
1506:
628:
615:
540:{\displaystyle k=A/{\mathfrak {m}}}
532:
493:{\displaystyle a_{1},\ldots ,a_{n}}
317:
289:
24:
1557:, so the ring has Krull dimension
837:
667:where the second dimension is the
169:
25:
2614:
2481:10.1090/s0002-9947-1947-0021694-1
2128:This justifies the definition of
2084:if and only if the local ring of
1734:{\displaystyle x+{\mathfrak {m}}}
1964:Serre's multiplicity conjectures
159:) if and only if the local ring
31:
2135:
1276:
297:{\displaystyle {\mathfrak {m}}}
115:Krull's principal ideal theorem
2487:
2456:
2419:
2390:
2381:
2349:
2340:
2286:
2254:
1864:
1861:
1829:
1826:
1794:
1778:
1541:
1528:
1523:
1517:
1458:
1452:
1446:
1433:
1428:
1422:
1395:
1382:
1377:
1371:
1324:
1311:
1303:
1297:
1215:is an indeterminate, the ring
1054:
1042:
1029:is regular if and only if the
357:
325:
13:
1:
2543:Lectures on Modules and Rings
2502:
2378:, p. 123, Theorem 11.22.
2362:
2199:global homological dimension
1960:Serre's inequality on height
1497:has exactly one prime ideal
1211:is the ring of integers and
502:regular system of parameters
7:
2376:Atiyah & Macdonald 1969
2316:
2099:unique factorization domain
1755:unique factorization domain
1751:AuslanderâBuchsbaum theorem
1330:{\displaystyle A=k/(x^{2})}
1108:
1025:of positive characteristic
906:of a Noetherian local ring
767:(i.e., the supremum of the
243:complete intersection rings
208:.) Regular local rings are
40:It has been suggested that
10:
2619:
2494:Is a regular ring a domain
2323:Geometrically regular ring
2139:
2113:is regular if and only if
1973:
1691:. (In fact it is equal to
1260:is a power series ring in
900:Multiplicity one criterion
442:{\displaystyle \dim A=n\,}
225:Universally catenary rings
43:Geometrically regular ring
2192:ring of regular functions
2178:lies in the fact that an
1870:{\displaystyle A\cong k]}
214:von Neumann regular rings
57:Proposed since July 2024.
2333:
2301:Hilbert's syzygy theorem
2186:(that is every point is
2142:von Neumann regular ring
1948:{\displaystyle d=\dim A}
996:transversal intersection
547:be the residue field of
2468:Trans. Amer. Math. Soc.
2174:The origin of the term
2076:. Zariski proved that
1970:Origin of basic notions
1955:, the Krull dimension.
1233:in the prime ideal (2,
1123:discrete valuation ring
204:is regular. (See also:
2293:
2239:principal ideal domain
2222:is regular then so is
1949:
1913:
1871:
1801:
1735:
1705:
1685:
1665:
1645:
1608:is the zero ideal, so
1602:
1571:
1551:
1491:
1468:
1331:
1256:that contains a field
1095:
1071:
1019:
972:
889:
845:
805:
785:
761:
737:
656:
581:
561:
541:
494:
443:
404:
384:
364:
298:
274:
190:
2294:
2190:) if and only if its
1950:
1914:
1872:
1802:
1736:
1706:
1686:
1666:
1646:
1603:
1572:
1552:
1492:
1469:
1332:
1096:
1072:
1020:
973:
890:
846:
806:
786:
769:projective dimensions
762:
738:
657:
582:
562:
542:
495:
444:
405:
385:
365:
299:
275:
191:
2248:
2000:contained in affine
1927:
1885:
1814:
1775:
1715:
1695:
1675:
1655:
1612:
1581:
1561:
1501:
1481:
1344:
1285:
1085:
1036:
1009:
918:
860:
821:
795:
775:
751:
680:
597:
571:
551:
513:
458:
420:
394:
374:
312:
284:
264:
231:CohenâMacaulay rings
163:
129:is regular whenever
50:into this article. (
2149:commutative algebra
2095:homological algebra
2008:, and suppose that
1671:dimension at least
1413:
1362:
1204:] is regular local.
1198:formal power series
1154:More generally, if
1135:formal power series
249:regular local rings
68:commutative algebra
2598:Algebraic geometry
2510:Atiyah, Michael F.
2414:10.1007/BF01160110
2309:A regular ring is
2289:
2080:is nonsingular at
2040:Jacobian condition
2030:is nonsingular at
1945:
1909:
1867:
1797:
1731:
1701:
1681:
1661:
1641:
1598:
1567:
1547:
1487:
1464:
1327:
1264:variables over an
1091:
1067:
1031:Frobenius morphism
1015:
968:
885:
868:
841:
829:
801:
781:
757:
733:
688:
652:
577:
557:
537:
500:are then called a
490:
439:
400:
380:
360:
294:
270:
186:
93:. In symbols, let
72:regular local ring
2561:Jean-Pierre Serre
2555:978-1-4612-0525-8
2514:Macdonald, Ian G.
2292:{\displaystyle k}
2233:In particular if
2205:Jean-Pierre Serre
2155:is a commutative
2107:Jean-Pierre Serre
1998:algebraic variety
1990:algebraic variety
1764:, as well as the
1704:{\displaystyle 1}
1684:{\displaystyle 1}
1664:{\displaystyle k}
1570:{\displaystyle 0}
1545:
1490:{\displaystyle A}
1450:
1414:
1399:
1363:
1242:structure theorem
1094:{\displaystyle R}
1018:{\displaystyle R}
965:
936:
867:
828:
804:{\displaystyle A}
784:{\displaystyle A}
760:{\displaystyle A}
728:
720:
719: is an
687:
580:{\displaystyle A}
560:{\displaystyle A}
403:{\displaystyle A}
383:{\displaystyle n}
273:{\displaystyle A}
256:Characterizations
146:algebraic variety
64:
63:
59:
16:(Redirected from
2610:
2532:
2496:
2491:
2485:
2484:
2483:
2460:
2454:
2453:
2423:
2417:
2416:
2394:
2388:
2385:
2379:
2373:
2356:
2353:
2347:
2344:
2298:
2296:
2295:
2290:
2285:
2284:
2266:
2265:
2236:
2216:Dedekind domains
2159:, such that the
2121:, i.e. if every
2119:global dimension
1954:
1952:
1951:
1946:
1918:
1916:
1915:
1910:
1908:
1907:
1901:
1876:
1874:
1873:
1868:
1860:
1859:
1841:
1840:
1806:
1804:
1803:
1798:
1793:
1792:
1745:Basic properties
1740:
1738:
1737:
1732:
1730:
1729:
1710:
1708:
1707:
1702:
1690:
1688:
1687:
1682:
1670:
1668:
1667:
1662:
1650:
1648:
1647:
1642:
1640:
1639:
1634:
1633:
1626:
1621:
1620:
1607:
1605:
1604:
1599:
1597:
1596:
1591:
1590:
1576:
1574:
1573:
1568:
1556:
1554:
1553:
1548:
1546:
1544:
1540:
1539:
1526:
1515:
1510:
1509:
1496:
1494:
1493:
1488:
1473:
1471:
1470:
1465:
1451:
1449:
1445:
1444:
1431:
1417:
1415:
1402:
1400:
1398:
1394:
1393:
1380:
1366:
1364:
1351:
1336:
1334:
1333:
1328:
1323:
1322:
1310:
1100:
1098:
1097:
1092:
1076:
1074:
1073:
1068:
1066:
1065:
1024:
1022:
1021:
1016:
1001:In the positive
977:
975:
974:
969:
967:
966:
958:
943:
938:
937:
929:
894:
892:
891:
886:
869:
865:
850:
848:
847:
842:
830:
826:
810:
808:
807:
802:
791:-modules.) Then
790:
788:
787:
782:
766:
764:
763:
758:
745:global dimension
742:
740:
739:
734:
729:
726:
721:
718:
689:
685:
661:
659:
658:
653:
638:
637:
632:
631:
624:
619:
618:
609:
608:
586:
584:
583:
578:
566:
564:
563:
558:
546:
544:
543:
538:
536:
535:
529:
499:
497:
496:
491:
489:
488:
470:
469:
448:
446:
445:
440:
409:
407:
406:
401:
389:
387:
386:
381:
369:
367:
366:
361:
356:
355:
337:
336:
321:
320:
303:
301:
300:
295:
293:
292:
279:
277:
276:
271:
237:Gorenstein rings
195:
193:
192:
187:
185:
184:
173:
172:
89:is equal to its
55:
35:
34:
27:
21:
2618:
2617:
2613:
2612:
2611:
2609:
2608:
2607:
2588:
2587:
2569:Springer-Verlag
2547:Springer-Verlag
2505:
2500:
2499:
2492:
2488:
2461:
2457:
2443:10.2307/2371447
2424:
2420:
2398:Krull, Wolfgang
2395:
2391:
2386:
2382:
2374:
2370:
2365:
2360:
2359:
2354:
2350:
2345:
2341:
2336:
2328:quasi-free ring
2319:
2280:
2276:
2261:
2257:
2249:
2246:
2245:
2243:polynomial ring
2234:
2169:Krull dimension
2157:Noetherian ring
2145:
2138:
2058:
2051:
2024:
2017:
1978:
1972:
1928:
1925:
1924:
1903:
1902:
1897:
1886:
1883:
1882:
1855:
1851:
1836:
1832:
1815:
1812:
1811:
1788:
1787:
1776:
1773:
1772:
1747:
1725:
1724:
1716:
1713:
1712:
1696:
1693:
1692:
1676:
1673:
1672:
1656:
1653:
1652:
1635:
1629:
1628:
1627:
1622:
1616:
1615:
1613:
1610:
1609:
1592:
1586:
1585:
1584:
1582:
1579:
1578:
1562:
1559:
1558:
1535:
1531:
1527:
1516:
1514:
1505:
1504:
1502:
1499:
1498:
1482:
1479:
1478:
1440:
1436:
1432:
1418:
1416:
1401:
1389:
1385:
1381:
1367:
1365:
1350:
1345:
1342:
1341:
1318:
1314:
1306:
1286:
1283:
1282:
1279:
1266:extension field
1226:(i.e. the ring
1225:
1180:
1171:
1164:
1158:is a field and
1149:p-adic integers
1129:is a field and
1111:
1086:
1083:
1082:
1061:
1057:
1037:
1034:
1033:
1010:
1007:
1006:
957:
956:
939:
928:
927:
919:
916:
915:
902:states: if the
863:
861:
858:
857:
856:in which case,
824:
822:
819:
818:
796:
793:
792:
776:
773:
772:
752:
749:
748:
725:
717:
683:
681:
678:
677:
669:Krull dimension
633:
627:
626:
625:
620:
614:
613:
604:
600:
598:
595:
594:
572:
569:
568:
552:
549:
548:
531:
530:
525:
514:
511:
510:
484:
480:
465:
461:
459:
456:
455:
421:
418:
417:
395:
392:
391:
375:
372:
371:
351:
347:
332:
328:
316:
315:
313:
310:
309:
288:
287:
285:
282:
281:
265:
262:
261:
258:
174:
168:
167:
166:
164:
161:
160:
112:
103:
91:Krull dimension
60:
36:
32:
23:
22:
15:
12:
11:
5:
2616:
2606:
2605:
2600:
2586:
2585:
2580:
2558:
2536:
2533:
2522:Addison-Wesley
2504:
2501:
2498:
2497:
2486:
2464:Zariski, Oscar
2455:
2431:Amer. J. Math.
2427:Zariski, Oscar
2418:
2389:
2380:
2367:
2366:
2364:
2361:
2358:
2357:
2348:
2338:
2337:
2335:
2332:
2331:
2330:
2325:
2318:
2315:
2288:
2283:
2279:
2275:
2272:
2269:
2264:
2260:
2256:
2253:
2180:affine variety
2137:
2134:
2056:
2049:
2022:
2015:
2004:-space over a
1982:Wolfgang Krull
1971:
1968:
1944:
1941:
1938:
1935:
1932:
1906:
1900:
1896:
1893:
1890:
1879:
1878:
1866:
1863:
1858:
1854:
1850:
1847:
1844:
1839:
1835:
1831:
1828:
1825:
1822:
1819:
1796:
1791:
1786:
1783:
1780:
1746:
1743:
1728:
1723:
1720:
1700:
1680:
1660:
1638:
1632:
1625:
1619:
1595:
1589:
1566:
1543:
1538:
1534:
1530:
1525:
1522:
1519:
1513:
1508:
1486:
1475:
1474:
1463:
1460:
1457:
1454:
1448:
1443:
1439:
1435:
1430:
1427:
1424:
1421:
1412:
1409:
1405:
1397:
1392:
1388:
1384:
1379:
1376:
1373:
1370:
1361:
1358:
1354:
1349:
1326:
1321:
1317:
1313:
1309:
1305:
1302:
1299:
1296:
1293:
1290:
1278:
1275:
1274:
1273:
1238:
1219:
1205:
1190:
1176:
1169:
1162:
1152:
1141:
1119:
1110:
1107:
1090:
1064:
1060:
1056:
1053:
1050:
1047:
1044:
1041:
1014:
1003:characteristic
964:
961:
955:
952:
949:
946:
942:
935:
932:
926:
923:
897:
896:
884:
881:
878:
875:
872:
854:
853:
852:
839:
836:
833:
813:
812:
800:
780:
756:
732:
724:
716:
713:
710:
707:
704:
701:
698:
695:
692:
673:
672:
665:
664:
663:
650:
647:
644:
641:
636:
630:
623:
617:
612:
607:
603:
589:
588:
576:
556:
534:
528:
524:
521:
518:
506:
505:
487:
483:
479:
476:
473:
468:
464:
452:
451:
450:
437:
434:
431:
428:
425:
412:
411:
399:
379:
359:
354:
350:
346:
343:
340:
335:
331:
327:
324:
319:
291:
269:
257:
254:
253:
252:
206:regular scheme
183:
180:
177:
171:
108:
101:
62:
61:
39:
37:
30:
9:
6:
4:
3:
2:
2615:
2604:
2601:
2599:
2596:
2595:
2593:
2584:
2581:
2579:. Chap.IV.D.
2578:
2577:3-540-66641-9
2574:
2570:
2566:
2565:Local algebra
2562:
2559:
2556:
2552:
2548:
2544:
2540:
2539:Tsit-Yuen Lam
2537:
2534:
2531:
2527:
2523:
2519:
2515:
2511:
2507:
2506:
2495:
2490:
2482:
2477:
2473:
2469:
2465:
2459:
2452:
2448:
2444:
2440:
2436:
2432:
2428:
2422:
2415:
2411:
2407:
2403:
2399:
2393:
2384:
2377:
2372:
2368:
2352:
2343:
2339:
2329:
2326:
2324:
2321:
2320:
2314:
2312:
2307:
2304:
2302:
2281:
2277:
2273:
2270:
2267:
2262:
2258:
2251:
2244:
2240:
2231:
2229:
2225:
2221:
2217:
2212:
2210:
2206:
2202:
2200:
2195:
2193:
2189:
2185:
2181:
2177:
2172:
2170:
2166:
2162:
2158:
2154:
2150:
2143:
2133:
2131:
2126:
2124:
2120:
2116:
2112:
2108:
2102:
2100:
2096:
2091:
2087:
2083:
2079:
2075:
2071:
2067:
2063:
2059:
2052:
2045:
2041:
2037:
2033:
2029:
2025:
2018:
2011:
2007:
2006:perfect field
2003:
1999:
1995:
1991:
1987:
1986:Oscar Zariski
1983:
1977:
1976:smooth scheme
1967:
1965:
1961:
1956:
1942:
1939:
1936:
1933:
1930:
1922:
1921:residue field
1898:
1894:
1891:
1888:
1856:
1852:
1848:
1845:
1842:
1837:
1833:
1823:
1820:
1817:
1810:
1809:
1808:
1784:
1781:
1769:
1767:
1763:
1758:
1756:
1752:
1742:
1741:is a basis.)
1721:
1718:
1698:
1678:
1658:
1636:
1623:
1593:
1564:
1536:
1532:
1520:
1511:
1484:
1461:
1455:
1441:
1437:
1425:
1419:
1410:
1407:
1403:
1390:
1386:
1374:
1368:
1359:
1356:
1352:
1347:
1340:
1339:
1338:
1319:
1315:
1307:
1300:
1294:
1291:
1288:
1271:
1267:
1263:
1259:
1255:
1251:
1247:
1243:
1239:
1236:
1232:
1229:
1223:
1218:
1214:
1210:
1206:
1203:
1199:
1195:
1191:
1188:
1184:
1179:
1175:
1168:
1161:
1157:
1153:
1150:
1146:
1142:
1139:
1136:
1132:
1128:
1124:
1120:
1117:
1113:
1112:
1106:
1104:
1088:
1080:
1062:
1058:
1051:
1048:
1045:
1039:
1032:
1028:
1012:
1004:
999:
997:
993:
989:
986:is one, then
985:
981:
978:) and if the
962:
959:
953:
950:
947:
944:
940:
933:
930:
924:
921:
913:
909:
905:
901:
882:
879:
876:
873:
870:
855:
834:
831:
817:
816:
815:
814:
811:is regular if
798:
778:
770:
754:
746:
722:
714:
711:
708:
705:
702:
693:
690:
675:
674:
670:
666:
648:
645:
642:
639:
634:
621:
610:
605:
601:
593:
592:
591:
590:
587:is regular if
574:
554:
526:
522:
519:
516:
508:
507:
503:
485:
481:
477:
474:
471:
466:
462:
453:
435:
432:
429:
426:
423:
416:
415:
414:
413:
410:is regular if
397:
377:
352:
348:
344:
341:
338:
333:
329:
322:
307:
306:
305:
267:
251:
250:
245:
244:
239:
238:
233:
232:
227:
226:
222:
221:
220:
217:
215:
211:
207:
203:
199:
181:
178:
175:
158:
154:
150:
147:
143:
138:
136:
132:
128:
124:
120:
117:implies that
116:
111:
107:
100:
96:
92:
88:
87:maximal ideal
84:
80:
77:
73:
69:
58:
53:
49:
45:
44:
38:
29:
28:
19:
2564:
2557:. Chap.5.G.
2542:
2517:
2489:
2471:
2467:
2458:
2434:
2430:
2421:
2405:
2401:
2392:
2383:
2371:
2351:
2342:
2308:
2305:
2232:
2227:
2223:
2219:
2213:
2208:
2203:
2196:
2194:is regular.
2176:regular ring
2175:
2173:
2161:localization
2153:regular ring
2152:
2146:
2136:Regular ring
2129:
2127:
2122:
2114:
2110:
2103:
2089:
2085:
2081:
2077:
2073:
2072:− dim
2069:
2065:
2061:
2054:
2047:
2043:
2038:satisfies a
2035:
2031:
2027:
2020:
2013:
2009:
2001:
1993:
1979:
1957:
1880:
1770:
1762:localization
1759:
1748:
1476:
1280:
1277:Non-examples
1269:
1261:
1257:
1253:
1234:
1227:
1221:
1216:
1212:
1208:
1201:
1193:
1186:
1182:
1177:
1173:
1166:
1159:
1155:
1144:
1137:
1130:
1126:
1026:
1000:
992:intersection
987:
983:
980:multiplicity
911:
907:
899:
898:
866:gl dim
827:gl dim
686:gl dim
501:
259:
248:
247:
241:
235:
229:
223:
218:
209:
201:
157:smooth point
148:
141:
139:
134:
130:
126:
122:
118:
109:
105:
98:
94:
71:
65:
56:
41:
18:Regular ring
2603:Ring theory
2437:: 187â221,
2408:: 745â766,
2241:, then the
2184:nonsingular
2165:prime ideal
2117:has finite
1246:Irvin Cohen
212:related to
153:nonsingular
2592:Categories
2503:References
2130:regularity
1974:See also:
1958:See also:
1766:completion
904:completion
83:generators
79:local ring
76:Noetherian
2363:Citations
2271:…
2163:at every
1940:
1846:…
1821:≅
1459:→
1453:→
1408:⋅
1357:⋅
1348:⋯
1281:The ring
1231:localized
1055:↦
1043:→
963:^
954:
934:^
925:
880:
838:∞
712:∣
706:
646:
611:
475:…
427:
342:…
2571:, 2000,
2549:, 1999,
2516:(1969),
2474:: 1â52,
2402:Math. Z.
2317:See also
1404:→
1353:→
1250:complete
1109:Examples
2530:0242802
2451:2371447
2311:reduced
2188:regular
1992:. Let
1919:is the
1240:By the
1172:, ...,
1103:reduced
771:of all
743:be the
727:-module
567:. Then
104:, ...,
85:of its
52:Discuss
2575:
2553:
2528:
2449:
2218:. If
2209:finite
1996:be an
1923:, and
1881:where
1760:Every
1711:since
1577:, but
1114:Every
370:where
144:on an
133:= dim
125:, and
121:â„ dim
48:merged
2447:JSTOR
2334:Notes
2042:: If
2019:,...,
1200:ring
1116:field
198:germs
74:is a
2573:ISBN
2551:ISBN
2151:, a
2046:= (â
1962:and
1749:The
1651:has
1248:, a
1220:(2,
1121:Any
1081:and
1079:flat
835:<
676:Let
509:Let
308:Let
70:, a
2476:doi
2439:doi
2410:doi
2230:.
2182:is
2147:In
2088:at
2068:is
2064:at
2034:if
2026:.
1937:dim
1771:If
1268:of
1244:of
1207:If
1192:If
1143:If
1101:is
1077:is
982:of
951:dim
922:dim
877:dim
747:of
697:sup
643:dim
602:dim
424:dim
228:â
210:not
200:at
196:of
155:(a
151:is
66:In
46:be
2594::
2567:,
2563:,
2545:,
2541:,
2526:MR
2524:,
2520:,
2512:;
2472:62
2470:,
2445:,
2435:62
2433:,
2406:42
2404:,
2303:.
2201:.
2171:.
2101:.
2053:/â
1966:.
1757:.
1165:,
998:.
914:,
703:pd
694::=
246:â
240:â
234:â
216:.
137:.
2478::
2441::
2412::
2287:]
2282:n
2278:X
2274:,
2268:,
2263:1
2259:X
2255:[
2252:k
2235:k
2228:A
2224:A
2220:A
2144:.
2123:A
2115:A
2111:A
2090:P
2086:Y
2082:P
2078:Y
2074:Y
2070:n
2066:P
2062:M
2057:j
2055:x
2050:i
2048:f
2044:M
2036:Y
2032:P
2028:Y
2023:m
2021:f
2016:1
2014:f
2010:Y
2002:n
1994:Y
1943:A
1934:=
1931:d
1905:m
1899:/
1895:A
1892:=
1889:k
1877:,
1865:]
1862:]
1857:d
1853:x
1849:,
1843:,
1838:1
1834:x
1830:[
1827:[
1824:k
1818:A
1795:)
1790:m
1785:,
1782:A
1779:(
1727:m
1722:+
1719:x
1699:1
1679:1
1659:k
1637:2
1631:m
1624:/
1618:m
1594:2
1588:m
1565:0
1542:)
1537:2
1533:x
1529:(
1524:)
1521:x
1518:(
1512:=
1507:m
1485:A
1462:0
1456:k
1447:)
1442:2
1438:x
1434:(
1429:]
1426:x
1423:[
1420:k
1411:x
1396:)
1391:2
1387:x
1383:(
1378:]
1375:x
1372:[
1369:k
1360:x
1325:)
1320:2
1316:x
1312:(
1308:/
1304:]
1301:x
1298:[
1295:k
1292:=
1289:A
1272:.
1270:k
1262:d
1258:k
1254:d
1235:X
1228:Z
1224:)
1222:X
1217:Z
1213:X
1209:Z
1202:A
1194:A
1189:.
1187:d
1183:k
1178:d
1174:X
1170:2
1167:X
1163:1
1160:X
1156:k
1145:p
1138:k
1131:X
1127:k
1089:R
1063:p
1059:r
1052:r
1049:,
1046:R
1040:R
1027:p
1013:R
988:A
984:A
960:A
948:=
945:p
941:/
931:A
912:p
908:A
895:.
883:A
874:=
871:A
851:,
832:A
799:A
779:A
755:A
731:}
723:A
715:M
709:M
700:{
691:A
671:.
662:,
649:A
640:=
635:2
629:m
622:/
616:m
606:k
575:A
555:A
533:m
527:/
523:A
520:=
517:k
504:.
486:n
482:a
478:,
472:,
467:1
463:a
449:,
436:n
433:=
430:A
398:A
378:n
358:)
353:n
349:a
345:,
339:,
334:1
330:a
326:(
323:=
318:m
290:m
268:A
202:x
182:x
179:,
176:X
170:O
149:X
142:x
135:A
131:n
127:A
123:A
119:n
110:n
106:a
102:1
99:a
95:A
54:)
20:)
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