603:
1619:, Interscience Tracts in Pure and Applied Mathematics, vol. 13 (reprint ed.), New York-London: Interscience Publishers a division of John Wiley & Sons, pp. xiii+234,
970:
854:
682:
514:
1067:
931:
1014:
898:
1435:"Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie"
656:
623:
534:
466:
346:
313:
714:
446:
391:
742:
1034:
815:
782:
762:
554:
486:
411:
280:
260:
240:
216:
193:
1221:
of this separable algebraic closure correspond to automorphisms of the corresponding strict
Henselization. For example, a strict Henselization of the field of
1473:
1658:
1624:
1434:
1249:
Every field is a
Henselian local ring. (But not every field with valuation is "Henselian" in the sense of the fourth definition above.)
1183:
489:
1423:
1684:
1418:
1213:
isomorphism. More precisely it depends on the choice of a separable algebraic closure of the residue field of
559:
1430:
1109:
1316:
1237:
1194:
satisfying an algebraic equation). This can be thought of as the "algebraic" part of the completion.
1137:
90:
will be assumed to be commutative, though there is also a theory of non-commutative
Henselian rings.
56:
936:
820:
661:
1187:
1177:
196:
172:
160:
134:
1454:
1645:, Lecture Notes in Mathematics, vol. 169, Berlin-New York: Springer-Verlag, pp. v+129,
1413:
794:
351:
A ring is called
Henselian if it is a direct product of a finite number of Henselian local rings.
87:
1039:
903:
975:
859:
1654:
1620:
1559:
1514:
1393:
1073:
494:
167:
44:
628:
608:
519:
451:
318:
285:
1646:
1594:
1578:
1549:
1533:
1504:
1488:
1446:
1383:
1281:
1069:
is an isomorphism. In fact, this property characterises
Henselian rings, resp. local rings.
687:
145:
114:
60:
1668:
1634:
1608:
1571:
1526:
1481:
1405:
605:. This follows from the fourth definition, and from the fact that for every K-automorphism
419:
364:
1664:
1630:
1604:
1567:
1522:
1477:
1401:
1253:
1165:
933:
is an isomorphism. This should be compared to the fact that for any
Zariski open covering
719:
1271:
1257:
1229:
1173:
1019:
800:
767:
747:
539:
471:
396:
265:
245:
225:
219:
201:
178:
1678:
1291:
1222:
156:
99:
1367:
1302:
1218:
1191:
1169:
716:. The converse of this assertion also holds, because for a normal field extension
1582:
1537:
1492:
1371:
1327:
1267:
1209:. The strict Henselization is not quite universal: it is unique, but only up to
1153:
1129:
52:
28:
1072:
Likewise strict
Henselian rings are the local rings of geometric points in the
17:
1554:
1509:
1388:
40:
1599:
1587:
Memoirs of the
College of Science, University of Kyoto. Series A: Mathematics
1563:
1518:
1397:
144:
A local ring is
Henselian if and only if every finite ring extension is a
1650:
1450:
1176:
then so is its
Henselization. For example, the Henselization of the
1125:
1358:, Springer monographs of mathematics, 2005, thm. 3.2.15, p. 69.
1228:
is given by the maximal unramified extension, generated by all
1263:
and rings of formal power series over a field, are Henselian.
1287:
The Henselization of a local ring is a Henselian local ring.
1277:
Rings of algebraic power series over a field are Henselian.
1197:
Similarly there is a strictly Henselian ring generated by
793:
Henselian rings are the local rings with respect to the
393:
is a Henselian field. Then every algebraic extension of
137:
monic polynomials can be lifted to a factorization in
1042:
1022:
978:
939:
906:
862:
823:
803:
770:
750:
722:
690:
664:
631:
611:
562:
542:
522:
497:
474:
454:
422:
399:
367:
321:
288:
268:
248:
228:
204:
181:
1472:, Mathematische Monographien, vol. II, Berlin:
55:. Azumaya originally allowed Henselian rings to be
1061:
1028:
1008:
964:
925:
892:
848:
809:
776:
756:
736:
708:
676:
650:
617:
597:
548:
528:
508:
480:
460:
440:
405:
385:
340:
307:
274:
254:
234:
210:
187:
1186:at the point (0,0,...) is the ring of algebraic
1116:to a Henselian ring can be extended uniquely to
66:Some standard references for Hensel rings are (
1266:The rings of convergent power series over the
413:is henselian (by the fourth definition above).
262:(resp. to every finite separable extension of
242:extends uniquely to every finite extension of
222:is Henselian. That is the case if and only if
1148:has the same completion and residue field as
109:if Hensel's lemma holds. This means that if
8:
959:
940:
843:
824:
75:
1470:Henselsche Ringe und algebraische Geometrie
1468:Kurke, H.; Pfister, G.; Roczen, M. (1975),
1301:is Henselian if and only if the associated
59:, but most authors now restrict them to be
1598:
1553:
1508:
1387:
1047:
1041:
1021:
977:
947:
938:
911:
905:
861:
831:
822:
802:
769:
749:
726:
721:
700:
695:
689:
663:
636:
630:
610:
561:
541:
521:
496:
473:
453:
421:
398:
366:
326:
320:
293:
287:
267:
247:
227:
203:
180:
1583:"On the theory of Henselian rings. III"
1474:VEB Deutscher Verlag der Wissenschaften
1347:
71:
48:
1538:"On the theory of Henselian rings. II"
1311:is Henselian (this is the quotient of
1105:
598:{\displaystyle v(\alpha ')=v(\alpha )}
121:, then any factorization of its image
67:
789:Henselian rings in algebraic geometry
7:
1439:Publications Mathématiques de l'IHÉS
1088:there is a universal Henselian ring
1284:over a Henselian ring is Henselian.
1136:is an algebraic substitute for the
1493:"On the theory of Henselian rings"
1236:. It is not "universal" as it has
25:
1294:of a Henselian ring is Henselian.
1256:local rings, such as the ring of
151:A Henselian local ring is called
1372:"On maximally central algebras."
817:is a Henselian local ring, and
218:is said to be Henselian if its
47:holds. They were introduced by
1053:
1003:
997:
965:{\displaystyle \{U_{i}\to X\}}
953:
917:
887:
881:
849:{\displaystyle \{U_{i}\to X\}}
837:
696:
677:{\displaystyle v\circ \sigma }
592:
586:
577:
566:
435:
423:
380:
368:
1:
1615:Nagata, Masayoshi (1975) ,
1542:Nagoya Mathematical Journal
1497:Nagoya Mathematical Journal
1419:Encyclopedia of Mathematics
1376:Nagoya Mathematical Journal
1330:then it is Henselian since
1317:ideal of nilpotent elements
856:is a Nisnevich covering of
784:are known to be conjugated.
1701:
1354:A. J. Engler, A. Prestel,
1062:{\displaystyle U_{i}\to X}
926:{\displaystyle U_{i}\to X}
1643:Anneaux locaux henséliens
1555:10.1017/s002776300001802x
1510:10.1017/s0027763000015439
1389:10.1017/s0027763000010114
1009:{\displaystyle X=Spec(R)}
893:{\displaystyle X=Spec(R)}
448:is a Henselian field and
1641:Raynaud, Michel (1970),
1412:Danilov, V. I. (2001) ,
509:{\displaystyle \alpha '}
1431:Grothendieck, Alexandre
1144:. The Henselization of
1132:. The Henselization of
1120:. The Henselization of
651:{\displaystyle K^{alg}}
618:{\displaystyle \sigma }
529:{\displaystyle \alpha }
461:{\displaystyle \alpha }
341:{\displaystyle K^{sep}}
308:{\displaystyle K^{alg}}
51:, who named them after
1600:10.1215/kjm/1250776700
1172:, normal, regular, or
1063:
1030:
1010:
966:
927:
894:
850:
811:
778:
758:
738:
710:
709:{\displaystyle v|_{K}}
678:
652:
619:
599:
550:
530:
510:
482:
462:
442:
407:
387:
342:
309:
276:
256:
236:
212:
189:
1280:A local ring that is
1064:
1031:
1011:
967:
928:
895:
851:
812:
797:in the sense that if
779:
759:
739:
711:
679:
653:
620:
600:
551:
531:
511:
483:
463:
443:
441:{\displaystyle (K,v)}
408:
388:
386:{\displaystyle (K,v)}
343:
310:
277:
257:
237:
213:
190:
1203:strict Henselization
1040:
1020:
976:
937:
904:
860:
821:
801:
768:
748:
744:, the extensions of
720:
688:
662:
629:
609:
560:
540:
520:
495:
472:
452:
420:
397:
365:
319:
286:
266:
246:
226:
202:
179:
168:abuse of terminology
133:) into a product of
1685:Commutative algebra
1188:formal power series
1178:ring of polynomials
1084:For any local ring
737:{\displaystyle L/K}
684:is an extension of
1651:10.1007/BFb0069571
1451:10.1007/BF02732123
1232:of order prime to
1110:local homomorphism
1059:
1026:
1006:
962:
923:
900:, then one of the
890:
846:
807:
795:Nisnevich topology
774:
754:
734:
706:
674:
648:
615:
595:
546:
526:
506:
478:
468:is algebraic over
458:
438:
403:
383:
338:
305:
272:
252:
232:
208:
185:
153:strictly Henselian
1660:978-3-540-05283-8
1626:978-0-88275-228-0
1579:Nagata, Masayoshi
1534:Nagata, Masayoshi
1489:Nagata, Masayoshi
1029:{\displaystyle R}
810:{\displaystyle R}
777:{\displaystyle L}
757:{\displaystyle v}
549:{\displaystyle K}
488:, then for every
481:{\displaystyle K}
406:{\displaystyle K}
275:{\displaystyle K}
255:{\displaystyle K}
235:{\displaystyle v}
211:{\displaystyle v}
188:{\displaystyle K}
76:Grothendieck 1967
70:, Chapter VII), (
16:(Redirected from
1692:
1671:
1637:
1611:
1602:
1574:
1557:
1529:
1512:
1484:
1464:
1463:
1462:
1453:, archived from
1426:
1408:
1391:
1359:
1352:
1108:, such that any
1104:, introduced by
1068:
1066:
1065:
1060:
1052:
1051:
1035:
1033:
1032:
1027:
1016:of a local ring
1015:
1013:
1012:
1007:
972:of the spectrum
971:
969:
968:
963:
952:
951:
932:
930:
929:
924:
916:
915:
899:
897:
896:
891:
855:
853:
852:
847:
836:
835:
816:
814:
813:
808:
783:
781:
780:
775:
763:
761:
760:
755:
743:
741:
740:
735:
730:
715:
713:
712:
707:
705:
704:
699:
683:
681:
680:
675:
657:
655:
654:
649:
647:
646:
624:
622:
621:
616:
604:
602:
601:
596:
576:
555:
553:
552:
547:
535:
533:
532:
527:
515:
513:
512:
507:
505:
487:
485:
484:
479:
467:
465:
464:
459:
447:
445:
444:
439:
412:
410:
409:
404:
392:
390:
389:
384:
347:
345:
344:
339:
337:
336:
314:
312:
311:
306:
304:
303:
281:
279:
278:
273:
261:
259:
258:
253:
241:
239:
238:
233:
217:
215:
214:
209:
194:
192:
191:
186:
161:separably closed
115:monic polynomial
86:In this article
21:
1700:
1699:
1695:
1694:
1693:
1691:
1690:
1689:
1675:
1674:
1661:
1640:
1627:
1614:
1577:
1532:
1487:
1467:
1460:
1458:
1429:
1411:
1366:
1363:
1362:
1353:
1349:
1344:
1336:
1310:
1272:complex numbers
1246:
1240:automorphisms.
1082:
1043:
1038:
1037:
1018:
1017:
974:
973:
943:
935:
934:
907:
902:
901:
858:
857:
827:
819:
818:
799:
798:
791:
766:
765:
746:
745:
718:
717:
694:
686:
685:
660:
659:
632:
627:
626:
607:
606:
569:
558:
557:
538:
537:
518:
517:
498:
493:
492:
470:
469:
450:
449:
418:
417:
395:
394:
363:
362:
358:
322:
317:
316:
289:
284:
283:
264:
263:
244:
243:
224:
223:
200:
199:
177:
176:
148:of local rings.
84:
78:, Chapter 18).
57:non-commutative
23:
22:
18:Henselian field
15:
12:
11:
5:
1698:
1696:
1688:
1687:
1677:
1676:
1673:
1672:
1659:
1638:
1625:
1612:
1575:
1530:
1485:
1465:
1427:
1409:
1361:
1360:
1346:
1345:
1343:
1340:
1339:
1338:
1334:
1320:
1308:
1295:
1288:
1285:
1278:
1275:
1274:are Henselian.
1264:
1261:-adic integers
1250:
1245:
1242:
1230:roots of unity
1081:
1078:
1074:Ă©tale topology
1058:
1055:
1050:
1046:
1025:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
961:
958:
955:
950:
946:
942:
922:
919:
914:
910:
889:
886:
883:
880:
877:
874:
871:
868:
865:
845:
842:
839:
834:
830:
826:
806:
790:
787:
786:
785:
773:
753:
733:
729:
725:
703:
698:
693:
673:
670:
667:
645:
642:
639:
635:
614:
594:
591:
588:
585:
582:
579:
575:
572:
568:
565:
545:
525:
504:
501:
477:
457:
437:
434:
431:
428:
425:
414:
402:
382:
379:
376:
373:
370:
357:
354:
353:
352:
349:
335:
332:
329:
325:
302:
299:
296:
292:
271:
251:
231:
220:valuation ring
207:
184:
164:
149:
142:
83:
80:
49:Azumaya (1951)
45:Hensel's lemma
33:Henselian ring
24:
14:
13:
10:
9:
6:
4:
3:
2:
1697:
1686:
1683:
1682:
1680:
1670:
1666:
1662:
1656:
1652:
1648:
1644:
1639:
1636:
1632:
1628:
1622:
1618:
1613:
1610:
1606:
1601:
1596:
1592:
1588:
1584:
1580:
1576:
1573:
1569:
1565:
1561:
1556:
1551:
1547:
1543:
1539:
1535:
1531:
1528:
1524:
1520:
1516:
1511:
1506:
1502:
1498:
1494:
1490:
1486:
1483:
1479:
1475:
1471:
1466:
1457:on 2016-03-03
1456:
1452:
1448:
1444:
1440:
1436:
1432:
1428:
1425:
1421:
1420:
1415:
1414:"Hensel ring"
1410:
1407:
1403:
1399:
1395:
1390:
1385:
1381:
1377:
1373:
1369:
1368:Azumaya, GorĂ´
1365:
1364:
1357:
1356:Valued fields
1351:
1348:
1341:
1333:
1329:
1326:has only one
1325:
1321:
1318:
1314:
1307:
1304:
1300:
1296:
1293:
1289:
1286:
1283:
1279:
1276:
1273:
1269:
1265:
1262:
1260:
1255:
1251:
1248:
1247:
1243:
1241:
1239:
1235:
1231:
1227:
1226:-adic numbers
1225:
1220:
1219:automorphisms
1216:
1212:
1208:
1204:
1201:, called the
1200:
1195:
1193:
1189:
1185:
1182:
1179:
1175:
1171:
1167:
1163:
1159:
1155:
1151:
1147:
1143:
1139:
1135:
1131:
1127:
1123:
1119:
1115:
1111:
1107:
1106:Nagata (1953)
1103:
1099:
1098:Henselization
1096:, called the
1095:
1092:generated by
1091:
1087:
1080:Henselization
1079:
1077:
1075:
1070:
1056:
1048:
1044:
1036:, one of the
1023:
1000:
994:
991:
988:
985:
982:
979:
956:
948:
944:
920:
912:
908:
884:
878:
875:
872:
869:
866:
863:
840:
832:
828:
804:
796:
788:
771:
751:
731:
727:
723:
701:
691:
671:
668:
665:
643:
640:
637:
633:
612:
589:
583:
580:
573:
570:
563:
543:
523:
502:
499:
491:
475:
455:
432:
429:
426:
415:
400:
377:
374:
371:
360:
359:
355:
350:
333:
330:
327:
323:
300:
297:
294:
290:
269:
249:
229:
221:
205:
198:
182:
175:
174:
169:
165:
162:
158:
157:residue field
154:
150:
147:
143:
140:
136:
132:
128:
124:
120:
116:
112:
108:
104:
101:
100:maximal ideal
97:
94:A local ring
93:
92:
91:
89:
81:
79:
77:
73:
69:
64:
62:
58:
54:
50:
46:
42:
38:
34:
30:
19:
1642:
1616:
1590:
1586:
1545:
1541:
1500:
1496:
1469:
1459:, retrieved
1455:the original
1442:
1438:
1417:
1379:
1375:
1355:
1350:
1331:
1323:
1312:
1305:
1303:reduced ring
1298:
1258:
1233:
1223:
1214:
1210:
1206:
1202:
1198:
1196:
1192:power series
1190:(the formal
1180:
1161:
1157:
1149:
1145:
1141:
1133:
1121:
1117:
1113:
1101:
1097:
1093:
1089:
1085:
1083:
1071:
792:
361:Assume that
171:
152:
138:
130:
126:
122:
118:
110:
106:
102:
95:
85:
72:Raynaud 1970
65:
36:
32:
26:
1617:Local rings
1382:: 119–150,
1337:is a field.
1328:prime ideal
1238:non-trivial
1154:flat module
1130:isomorphism
315:, resp. to
282:, resp. to
82:Definitions
68:Nagata 1975
61:commutative
53:Kurt Hensel
37:Hensel ring
29:mathematics
1593:: 93–101,
1461:2007-12-09
1342:References
1211:non-unique
1166:Noetherian
1138:completion
1124:is unique
356:Properties
105:is called
41:local ring
1564:0027-7630
1519:0027-7630
1503:: 45–57,
1445:: 5–361,
1424:EMS Press
1398:0027-7630
1254:Hausdorff
1252:Complete
1184:localized
1174:excellent
1152:and is a
1054:→
954:→
918:→
838:→
672:σ
669:∘
613:σ
590:α
571:α
524:α
500:α
490:conjugate
456:α
197:valuation
107:Henselian
43:in which
1679:Category
1581:(1959),
1548:: 1–19,
1536:(1954),
1491:(1953),
1433:(1967),
1370:(1951),
1292:quotient
1282:integral
1244:Examples
574:′
503:′
74:), and (
39:) is a
1669:0277519
1635:0155856
1609:0109835
1572:0067865
1527:0051821
1482:0491694
1406:0040287
1315:by the
1297:A ring
1170:reduced
1128:unique
155:if its
146:product
135:coprime
1667:
1657:
1633:
1623:
1607:
1570:
1562:
1525:
1517:
1480:
1404:
1396:
1290:Every
1217:, and
1160:. If
1156:over
1126:up to
1112:from
536:over
195:with
173:field
113:is a
98:with
88:rings
1655:ISBN
1621:ISBN
1560:ISSN
1515:ISSN
1394:ISSN
1268:real
170:, a
125:in (
35:(or
31:, a
1647:doi
1595:doi
1550:doi
1505:doi
1447:doi
1384:doi
1335:red
1322:If
1309:red
1270:or
1205:of
1164:is
1140:of
1100:of
764:to
625:of
516:of
416:If
166:By
159:is
117:in
27:In
1681::
1665:MR
1663:,
1653:,
1631:MR
1629:,
1605:MR
1603:,
1591:32
1589:,
1585:,
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1566:,
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1544:,
1540:,
1523:MR
1521:,
1513:,
1499:,
1495:,
1478:MR
1476:,
1443:32
1441:,
1437:,
1422:,
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1402:MR
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1168:,
1076:.
658:,
556:,
348:).
63:.
1649::
1597::
1552::
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1449::
1386::
1380:2
1332:A
1324:A
1313:A
1306:A
1299:A
1259:p
1234:p
1224:p
1215:A
1207:A
1199:A
1181:k
1162:A
1158:A
1150:A
1146:A
1142:A
1134:A
1122:A
1118:B
1114:A
1102:A
1094:A
1090:B
1086:A
1057:X
1049:i
1045:U
1024:R
1004:)
1001:R
998:(
995:c
992:e
989:p
986:S
983:=
980:X
960:}
957:X
949:i
945:U
941:{
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909:U
888:)
885:R
882:(
879:c
876:e
873:p
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864:X
844:}
841:X
833:i
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825:{
805:R
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752:v
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641:l
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587:(
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567:(
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433:v
430:,
427:K
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378:v
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291:K
270:K
250:K
230:v
206:v
183:K
163:.
141:.
139:R
131:m
129:/
127:R
123:P
119:R
111:P
103:m
96:R
20:)
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