1348:, and this seems to have been Grothendieck's motivation for defining them. Grothendieck (1965) observed that if it is possible to resolve singularities of all complete integral local Noetherian rings, then it is possible to resolve the singularities of all reduced quasi-excellent rings. Hironaka (1964)
1352:
this for all complete integral
Noetherian local rings over a field of characteristic 0, which implies his theorem that all singularities of excellent schemes over a field of characteristic 0 can be resolved. Conversely if it is possible to resolve all singularities of the spectra of all integral
804:
has the same length. This is useful for studying the dimension theory of such rings because their dimension can be bounded by a fixed maximal chain. In practice, this means infinite-dimensional
Noetherian rings which have an inductive definition of maximal chains of prime ideals, giving an
90:
0, but the positive characteristic case is (as of 2024) still a major open problem. Essentially all
Noetherian rings that occur naturally in algebraic geometry or number theory are excellent; in fact it is quite hard to construct examples of Noetherian rings that are not excellent.
1320:
has a cusp singularity at every closed point, so the set of singular points is not closed, though it is a G-ring. This ring is also universally catenary, as its localization at every prime ideal is a quotient of a regular ring.
99:
The definition of excellent rings is quite involved, so we recall the definitions of the technical conditions it satisfies. Although it seems like a long list of conditions, most rings in practice are excellent, such as
719:
604:
420:
551:
374:
453:
1104:
477:
868:
768:
is called excellent or quasi-excellent if it has a cover by open affine subschemes with the same property, which implies that every open affine subscheme has this property.
292:
136:
893:
329:
1124:
831:
794:
747:
656:
636:
500:
259:
239:
219:
193:
173:
1504:
1332:
of a 2-dimensional
Noetherian local ring that is catenary but not universally catenary is a G-ring, and is also a J-2 ring as any local G-ring is a J-2 ring (
59:
and others found several strange counterexamples showing that in general
Noetherian rings need not be well-behaved: for example, a normal Noetherian
761:. In practice almost all Noetherian rings are universally catenary, so there is little difference between excellent and quasi-excellent rings.
665:
143:
556:
1615:
379:
1522:
1569:
Hironaka, Heisuke (1964). "Resolution of
Singularities of an Algebraic Variety Over a Field of Characteristic Zero: II".
515:
338:
1639:
1634:
1517:
1370:
1345:
79:
1448:"ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique : IV. Ătude locale des schĂ©mas et des morphismes de schĂ©mas, Seconde partie"
938:
Most naturally occurring commutative rings in number theory or algebraic geometry are excellent. In particular:
87:
425:
1014:
1136:
1491:
1011:
Any finitely generated algebra over an excellent ring is excellent. This includes all polynomial algebras
71:
197:
1532:(1964). "Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I".
915:
758:
458:
40:
32:
1512:
264:
1290:
922:
101:
64:
55:. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but
17:
119:
1594:
1557:
1349:
1241:
52:
44:
1261:
Here is an example of a ring that is a G-ring but not a J-2 ring and so not quasi-excellent. If
512:) if it is Noetherian and its formal fibers are geometrically regular; this means that for any
836:
1611:
1586:
1549:
1495:
987:
302:
43:. Excellent rings are one answer to the problem of finding a natural class of "well-behaved"
1578:
1541:
1529:
308:
56:
28:
797:
113:
105:
25:
1277:
in infinitely many generators generated by the squares and cubes of all generators, and
873:
74:(1965) as a candidate for such a class of well-behaved rings. Quasi-excellent rings are
1109:
954:
816:
779:
732:
641:
621:
485:
244:
224:
204:
178:
158:
1628:
1499:
1329:
139:
109:
48:
1336:, p.88, 260). So it is a quasi-excellent catenary local ring that is not excellent.
1470:"Section 108.14 (02JC): A Noetherian ring of infinite dimensionâThe Stacks project"
1447:
990:
911:
295:
1126:
excellent. This means most rings considered in algebraic geometry are excellent.
904:
801:
75:
60:
1590:
1553:
1469:
1417:
1393:
979:
is excellent. Dedekind domains over fields of characteristic greater than
942:
All complete
Noetherian local rings, for instance all fields and the ring
800:
by definition. Because it is universally catenary, every maximal chain of
611:
1228:
rings. It is also universally catenary as it is a
Dedekind domain. Here
1598:
1561:
1268:
976:
714:{\displaystyle {\text{Reg}}({\text{Spec}}(S))\subset {\text{Spec}}(S)}
504:
1582:
1545:
599:{\displaystyle R_{\mathfrak {p}}\to {\hat {R_{\mathfrak {p}}}}}
1418:"Section 15.46 (07P6): The singular locusâThe Stacks project"
1344:
Quasi-excellent rings are closely related to the problem of
1388:
1386:
1289:
by adjoining inverses to all elements not in any of the
1220:
ring as all
Noetherian local rings of dimension at most
1112:
1017:
876:
839:
819:
782:
735:
668:
644:
624:
559:
518:
488:
461:
428:
415:{\displaystyle S\otimes _{R}\kappa ({\mathfrak {p}})}
382:
341:
311:
267:
247:
227:
207:
181:
161:
122:
1310:is a 1-dimensional Noetherian domain that is not a
1008:
Any localization of an excellent ring is excellent.
546:{\displaystyle {\mathfrak {p}}\in {\text{Spec}}(R)}
369:{\displaystyle {\mathfrak {p}}\in {\text{Spec}}(R)}
31:that behaves well with respect to the operation of
1394:"Section 15.49 (07GG): G-ringsâThe Stacks project"
1118:
1098:
887:
862:
825:
805:infinite-dimensional ring, cannot be constructed.
788:
741:
713:
650:
630:
598:
545:
494:
471:
447:
414:
368:
323:
286:
253:
233:
213:
187:
167:
130:
1200:such that is finite then the formal fibers of
606:to its completion is regular in the sense above.
422:is geometrically regular over the residue field
78:to be the base rings for which the problem of
1610:. Reading, Mass.: Benjamin/Cummings Pub. Co.
753:if it is a G-ring and J-2 ring. It is called
8:
1325:A quasi-excellent ring that is not excellent
70:The class of excellent rings was defined by
47:containing most of the rings that occur in
1606:Matsumura, Hideyuki (1980). "Chapter 13".
1333:
1111:
1087:
1068:
1056:
1047:
1028:
1016:
921:Any quasi-excellent normal local ring is
875:
838:
818:
781:
734:
697:
677:
669:
667:
643:
623:
583:
582:
576:
575:
565:
564:
558:
529:
520:
519:
517:
487:
463:
462:
460:
436:
435:
427:
403:
402:
390:
381:
352:
343:
342:
340:
310:
275:
266:
246:
226:
206:
180:
160:
124:
123:
121:
1163:-ring and so is not quasi-excellent. If
448:{\displaystyle \kappa ({\mathfrak {p}})}
83:
1382:
1353:finite algebras over a Noetherian ring
1099:{\displaystyle R/(f_{1},\ldots ,f_{k})}
965:All Dedekind domains of characteristic
969:are excellent. In particular the ring
1206:are not all geometrically regular so
993:in a finite number of variables over
7:
1505:Publications MathĂ©matiques de l'IHĂS
1452:Publications MathĂ©matiques de l'IHĂS
1441:
1439:
1437:
1501:Eléments de géométrie algébrique IV
833:and a locally finite type morphism
584:
566:
521:
464:
437:
404:
344:
14:
1446:Grothendieck, Alexander (1965).
725:Definition of (quasi-)excellence
662:, meaning the regular subscheme
1257:A G-ring that is not a J-2 ring
1169:is any field of characteristic
1131:A J-2 ring that is not a G-ring
472:{\displaystyle {\mathfrak {p}}}
116:over characteristic 0 (such as
1093:
1061:
1053:
1021:
903:Any quasi-excellent ring is a
854:
708:
702:
691:
688:
682:
674:
590:
572:
553:, the map from the local ring
540:
534:
442:
432:
409:
399:
363:
357:
315:
287:{\displaystyle R\otimes _{k}K}
1:
757:if it is quasi-excellent and
1185:is the ring of power series
335:if it is flat and for every
221:if for any finite extension
131:{\displaystyle \mathbb {Z} }
1518:Encyclopedia of Mathematics
1371:Resolution of singularities
1346:resolution of singularities
1340:Resolution of singularities
80:resolution of singularities
1656:
813:Given an excellent scheme
776:Because an excellent ring
863:{\displaystyle f:X'\to X}
1474:stacks.math.columbia.edu
1422:stacks.math.columbia.edu
1398:stacks.math.columbia.edu
1135:Here is an example of a
1271:of the polynomial ring
1137:discrete valuation ring
1511:V.I. Danilov (2001) ,
1492:Alexandre Grothendieck
1120:
1100:
983:need not be excellent.
889:
864:
827:
790:
743:
715:
652:
632:
600:
547:
496:
473:
449:
416:
370:
325:
324:{\displaystyle R\to S}
288:
255:
235:
215:
189:
169:
146:rings of these rings.
132:
72:Alexander Grothendieck
1571:Annals of Mathematics
1534:Annals of Mathematics
1234:denotes the image of
1121:
1101:
890:
865:
828:
791:
744:
716:
653:
633:
601:
548:
497:
474:
450:
417:
371:
326:
289:
256:
236:
216:
198:geometrically regular
190:
170:
133:
1507:24 (1965), section 7
1361:is quasi-excellent.
1110:
1015:
916:analytically reduced
910:Any quasi-excellent
874:
837:
817:
780:
759:universally catenary
733:
666:
642:
622:
557:
516:
486:
459:
426:
380:
339:
309:
265:
245:
225:
205:
179:
159:
150:Recalled definitions
120:
41:universally catenary
22:quasi-excellent ring
1640:Commutative algebra
1608:Commutative algebra
1148:and characteristic
923:analytically normal
796:is a G-ring, it is
618:if any finite type
610:Finally, a ring is
175:containing a field
65:analytically normal
35:, and is called an
18:commutative algebra
1635:Algebraic geometry
1293:generated by some
1242:Frobenius morphism
1116:
1096:
888:{\displaystyle X'}
885:
860:
823:
786:
739:
711:
648:
628:
596:
543:
492:
469:
445:
412:
366:
321:
284:
251:
231:
211:
185:
165:
128:
112:Noetherian rings,
53:algebraic geometry
1530:Hironaka, Heisuke
1283:is obtained from
1119:{\displaystyle R}
826:{\displaystyle X}
789:{\displaystyle R}
742:{\displaystyle R}
700:
680:
672:
651:{\displaystyle S}
631:{\displaystyle R}
593:
532:
510:Grothendieck ring
495:{\displaystyle R}
355:
254:{\displaystyle k}
234:{\displaystyle K}
214:{\displaystyle k}
188:{\displaystyle k}
168:{\displaystyle R}
1647:
1621:
1602:
1565:
1525:
1513:"Excellent ring"
1484:
1483:
1481:
1480:
1466:
1460:
1459:
1443:
1432:
1431:
1429:
1428:
1414:
1408:
1407:
1405:
1404:
1390:
1330:Nagata's example
1319:
1313:
1309:
1303:
1288:
1282:
1276:
1266:
1252:
1239:
1233:
1227:
1223:
1219:
1215:
1211:
1205:
1199:
1184:
1178:
1174:
1168:
1162:
1158:
1154:
1147:
1143:
1125:
1123:
1122:
1117:
1105:
1103:
1102:
1097:
1092:
1091:
1073:
1072:
1060:
1052:
1051:
1033:
1032:
1004:
998:
982:
974:
968:
962:, are excellent.
959:
952:
899:Quasi-excellence
894:
892:
891:
886:
884:
869:
867:
866:
861:
853:
832:
830:
829:
824:
795:
793:
792:
787:
748:
746:
745:
740:
720:
718:
717:
712:
701:
698:
681:
678:
673:
670:
657:
655:
654:
649:
637:
635:
634:
629:
605:
603:
602:
597:
595:
594:
589:
588:
587:
577:
571:
570:
569:
552:
550:
549:
544:
533:
530:
525:
524:
501:
499:
498:
493:
478:
476:
475:
470:
468:
467:
454:
452:
451:
446:
441:
440:
421:
419:
418:
413:
408:
407:
395:
394:
375:
373:
372:
367:
356:
353:
348:
347:
330:
328:
327:
322:
293:
291:
290:
285:
280:
279:
260:
258:
257:
252:
240:
238:
237:
232:
220:
218:
217:
212:
194:
192:
191:
186:
174:
172:
171:
166:
137:
135:
134:
129:
127:
114:Dedekind domains
106:polynomial rings
57:Masayoshi Nagata
29:commutative ring
1655:
1654:
1650:
1649:
1648:
1646:
1645:
1644:
1625:
1624:
1618:
1605:
1583:10.2307/1970547
1568:
1546:10.2307/1970486
1528:
1510:
1488:
1487:
1478:
1476:
1468:
1467:
1463:
1445:
1444:
1435:
1426:
1424:
1416:
1415:
1411:
1402:
1400:
1392:
1391:
1384:
1379:
1367:
1342:
1327:
1315:
1311:
1305:
1302:
1294:
1284:
1278:
1272:
1262:
1259:
1244:
1235:
1229:
1225:
1221:
1217:
1216:-ring. It is a
1213:
1207:
1201:
1195:
1186:
1180:
1176:
1170:
1164:
1160:
1156:
1149:
1145:
1139:
1133:
1108:
1107:
1083:
1064:
1043:
1024:
1013:
1012:
1000:
994:
980:
970:
966:
955:
951:
943:
936:
934:Excellent rings
931:
901:
877:
872:
871:
846:
835:
834:
815:
814:
811:
778:
777:
774:
751:quasi-excellent
731:
730:
727:
664:
663:
640:
639:
620:
619:
578:
560:
555:
554:
514:
513:
484:
483:
457:
456:
424:
423:
386:
378:
377:
337:
336:
307:
306:
271:
263:
262:
243:
242:
223:
222:
203:
202:
177:
176:
157:
156:
152:
118:
117:
97:
86:showed this in
84:Hironaka (1964)
82:can be solved;
12:
11:
5:
1653:
1651:
1643:
1642:
1637:
1627:
1626:
1623:
1622:
1616:
1603:
1577:(2): 205â326.
1566:
1540:(1): 109â203.
1526:
1508:
1496:Jean Dieudonné
1486:
1485:
1461:
1433:
1409:
1381:
1380:
1378:
1375:
1374:
1373:
1366:
1363:
1357:then the ring
1341:
1338:
1334:Matsumura 1980
1326:
1323:
1298:
1258:
1255:
1191:
1132:
1129:
1128:
1127:
1115:
1095:
1090:
1086:
1082:
1079:
1076:
1071:
1067:
1063:
1059:
1055:
1050:
1046:
1042:
1039:
1036:
1031:
1027:
1023:
1020:
1009:
1006:
1005:are excellent.
984:
963:
960:-adic integers
947:
935:
932:
930:
927:
914:local ring is
900:
897:
895:is excellent.
883:
880:
859:
856:
852:
849:
845:
842:
822:
810:
807:
785:
773:
770:
738:
726:
723:
710:
707:
704:
696:
693:
690:
687:
684:
676:
647:
627:
608:
607:
592:
586:
581:
574:
568:
563:
542:
539:
536:
528:
523:
491:
480:
466:
444:
439:
434:
431:
411:
406:
401:
398:
393:
389:
385:
365:
362:
359:
351:
346:
320:
317:
314:
305:of rings from
299:
283:
278:
274:
270:
250:
230:
210:
184:
164:
151:
148:
126:
96:
93:
88:characteristic
39:if it is also
37:excellent ring
13:
10:
9:
6:
4:
3:
2:
1652:
1641:
1638:
1636:
1633:
1632:
1630:
1619:
1617:0-8053-7026-9
1613:
1609:
1604:
1600:
1596:
1592:
1588:
1584:
1580:
1576:
1572:
1567:
1563:
1559:
1555:
1551:
1547:
1543:
1539:
1535:
1531:
1527:
1524:
1520:
1519:
1514:
1509:
1506:
1503:
1502:
1497:
1493:
1490:
1489:
1475:
1471:
1465:
1462:
1457:
1453:
1449:
1442:
1440:
1438:
1434:
1423:
1419:
1413:
1410:
1399:
1395:
1389:
1387:
1383:
1376:
1372:
1369:
1368:
1364:
1362:
1360:
1356:
1351:
1347:
1339:
1337:
1335:
1331:
1324:
1322:
1318:
1308:
1301:
1297:
1292:
1287:
1281:
1275:
1270:
1265:
1256:
1254:
1251:
1247:
1243:
1238:
1232:
1210:
1204:
1198:
1194:
1190:
1183:
1173:
1167:
1152:
1144:of dimension
1142:
1138:
1130:
1113:
1088:
1084:
1080:
1077:
1074:
1069:
1065:
1057:
1048:
1044:
1040:
1037:
1034:
1029:
1025:
1018:
1010:
1007:
1003:
997:
992:
989:
986:The rings of
985:
978:
973:
964:
961:
958:
950:
946:
941:
940:
939:
933:
928:
926:
924:
919:
917:
913:
908:
906:
898:
896:
881:
878:
857:
850:
847:
843:
840:
820:
808:
806:
803:
799:
783:
771:
769:
767:
762:
760:
756:
752:
736:
724:
722:
705:
694:
685:
661:
645:
625:
617:
615:
614:
579:
561:
537:
526:
511:
507:
506:
489:
481:
429:
396:
391:
387:
383:
360:
349:
334:
318:
312:
304:
300:
297:
281:
276:
272:
268:
248:
228:
208:
200:
199:
182:
162:
154:
153:
149:
147:
145:
141:
115:
111:
107:
103:
94:
92:
89:
85:
81:
77:
73:
68:
66:
62:
58:
54:
50:
49:number theory
46:
42:
38:
34:
30:
27:
23:
19:
1607:
1574:
1570:
1537:
1533:
1516:
1500:
1477:. Retrieved
1473:
1464:
1455:
1451:
1425:. Retrieved
1421:
1412:
1401:. Retrieved
1397:
1358:
1354:
1343:
1328:
1316:
1306:
1299:
1295:
1285:
1279:
1273:
1263:
1260:
1249:
1245:
1236:
1230:
1208:
1202:
1196:
1192:
1188:
1181:
1171:
1165:
1150:
1140:
1134:
1001:
995:
991:power series
971:
956:
948:
944:
937:
920:
909:
902:
812:
802:prime ideals
775:
765:
763:
754:
750:
728:
659:
616:
612:
609:
509:
503:
502:is called a
332:
303:homomorphism
196:
144:localization
98:
69:
63:need not be
36:
21:
15:
905:Nagata ring
95:Definitions
76:conjectured
1629:Categories
1479:2020-07-24
1427:2020-07-24
1403:2020-07-24
1377:References
1240:under the
1159:but not a
988:convergent
798:Noetherian
772:Properties
749:is called
376:the fiber
331:is called
195:is called
61:local ring
33:completion
26:Noetherian
1591:0003-486X
1554:0003-486X
1523:EMS Press
1212:is not a
1155:which is
1078:…
1038:…
855:→
755:excellent
721:is open.
695:⊂
638:-algebra
591:^
573:→
527:∈
430:κ
397:κ
388:⊗
350:∈
316:→
273:⊗
261:the ring
1458:: 5â231.
1365:See also
1314:ring as
977:integers
929:Examples
882:′
851:′
140:quotient
110:complete
1599:1970547
1562:1970486
1304:, then
1269:subring
1267:is the
912:reduced
870:, then
809:Schemes
729:A ring
482:A ring
333:regular
296:regular
155:A ring
138:), and
1614:
1597:
1589:
1560:
1552:
1350:proved
1291:ideals
1153:> 0
766:scheme
505:G-ring
102:fields
1595:JSTOR
1558:JSTOR
1175:with
1106:with
201:over
45:rings
24:is a
1612:ISBN
1587:ISSN
1550:ISSN
1224:are
1179:and
699:Spec
679:Spec
531:Spec
508:(or
354:Spec
142:and
67:.
51:and
20:, a
1579:doi
1542:doi
1312:J-1
1226:J-2
1218:J-2
1177:= â
1157:J-2
999:or
975:of
953:of
671:Reg
660:J-1
658:is
613:J-2
455:of
294:is
241:of
16:In
1631::
1593:.
1585:.
1575:79
1573:.
1556:.
1548:.
1538:79
1536:.
1521:,
1515:,
1498:,
1494:,
1472:.
1456:24
1454:.
1450:.
1436:^
1420:.
1396:.
1385:^
1253:.
1248:â
925:.
918:.
907:.
764:A
301:A
108:,
104:,
1620:.
1601:.
1581::
1564:.
1544::
1482:.
1430:.
1406:.
1359:R
1355:R
1317:S
1307:S
1300:n
1296:x
1286:R
1280:S
1274:k
1264:R
1250:a
1246:a
1237:k
1231:k
1222:1
1214:G
1209:A
1203:A
1197:x
1193:i
1189:a
1187:ÎŁ
1182:A
1172:p
1166:k
1161:G
1151:p
1146:1
1141:A
1114:R
1094:)
1089:k
1085:f
1081:,
1075:,
1070:1
1066:f
1062:(
1058:/
1054:]
1049:n
1045:x
1041:,
1035:,
1030:1
1026:x
1022:[
1019:R
1002:C
996:R
981:0
972:Z
967:0
957:p
949:p
945:Z
879:X
858:X
848:X
844::
841:f
821:X
784:R
737:R
709:)
706:S
703:(
692:)
689:)
686:S
683:(
675:(
646:S
626:R
585:p
580:R
567:p
562:R
541:)
538:R
535:(
522:p
490:R
479:.
465:p
443:)
438:p
433:(
410:)
405:p
400:(
392:R
384:S
364:)
361:R
358:(
345:p
319:S
313:R
298:.
282:K
277:k
269:R
249:k
229:K
209:k
183:k
163:R
125:Z
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.