592:
1608:, 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,
959:
843:
671:
503:
1056:
920:
1003:
887:
1424:"Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie"
645:
612:
523:
455:
335:
302:
703:
435:
380:
731:
1023:
804:
771:
751:
543:
475:
400:
269:
249:
229:
205:
182:
1210:
of this separable algebraic closure correspond to automorphisms of the corresponding strict
Henselization. For example, a strict Henselization of the field of
1098:
1423:
1238:
Every field is a
Henselian local ring. (But not every field with valuation is "Henselian" in the sense of the fourth definition above.)
1462:
1647:
1613:
1172:
478:
1202:
isomorphism. More precisely it depends on the choice of a separable algebraic closure of the residue field of
1412:
548:
1673:
1407:
1183:
satisfying an algebraic equation). This can be thought of as the "algebraic" part of the completion.
79:
will be assumed to be commutative, though there is also a theory of non-commutative
Henselian rings.
925:
809:
650:
1443:
1634:, Lecture Notes in Mathematics, vol. 169, Berlin-New York: Springer-Verlag, pp. v+129,
1402:
340:
A ring is called
Henselian if it is a direct product of a finite number of Henselian local rings.
1028:
892:
1419:
964:
848:
483:
617:
597:
508:
440:
307:
274:
1058:
is an isomorphism. In fact, this property characterises
Henselian rings, resp. local rings.
676:
1657:
1623:
1597:
1560:
1515:
1470:
1394:
1305:
1226:
1126:
594:. This follows from the fourth definition, and from the fact that for every K-automorphism
408:
353:
45:
922:
is an isomorphism. This should be compared to the fact that for any
Zariski open covering
8:
1176:
1166:
708:
185:
161:
149:
123:
1008:
789:
783:
756:
736:
528:
460:
385:
254:
234:
214:
190:
167:
76:
1643:
1609:
1548:
1503:
1382:
156:
33:
1062:
1635:
1583:
1567:
1538:
1522:
1493:
1477:
1435:
1372:
1270:
134:
103:
49:
705:. The converse of this assertion also holds, because for a normal field extension
1653:
1619:
1593:
1571:
1556:
1526:
1511:
1481:
1466:
1390:
1360:
1242:
1154:
1198:. The strict Henselization is not quite universal: it is unique, but only up to
1260:
1246:
1218:
1162:
1061:
Likewise strict
Henselian rings are the local rings of geometric points in the
208:
1543:
1498:
1377:
1667:
1588:
1576:
Memoirs of the
College of Science, University of Kyoto. Series A: Mathematics
1552:
1507:
1386:
1280:
1211:
145:
88:
1356:
1291:
1207:
1180:
1158:
133:
A local ring is
Henselian if and only if every finite ring extension is a
1316:
1256:
1142:
1118:
41:
17:
1639:
1439:
29:
1165:
then so is its
Henselization. For example, the Henselization of the
1347:, Springer monographs of mathematics, 2005, thm. 3.2.15, p. 69.
1217:
is given by the maximal unramified extension, generated by all
1252:
and rings of formal power series over a field, are Henselian.
1114:
1276:
The Henselization of a local ring is a Henselian local ring.
1266:
Rings of algebraic power series over a field are Henselian.
1186:
Similarly there is a strictly Henselian ring generated by
782:
Henselian rings are the local rings with respect to the
777:
382:
is a Henselian field. Then every algebraic extension of
126:
monic polynomials can be lifted to a factorization in
1031:
1011:
967:
928:
895:
851:
812:
792:
759:
739:
711:
679:
653:
620:
600:
551:
531:
511:
486:
463:
443:
411:
388:
356:
310:
277:
257:
237:
217:
193:
170:
1461:, Mathematische Monographien, vol. II, Berlin:
44:. Azumaya originally allowed Henselian rings to be
1456:
1050:
1017:
997:
953:
914:
881:
837:
798:
765:
745:
725:
697:
665:
639:
606:
586:
537:
517:
497:
469:
449:
429:
394:
374:
329:
296:
263:
243:
223:
199:
176:
1665:
1175:at the point (0,0,...) is the ring of algebraic
1105:to a Henselian ring can be extended uniquely to
55:Some standard references for Hensel rings are (
1255:The rings of convergent power series over the
402:is henselian (by the fourth definition above).
251:(resp. to every finite separable extension of
231:extends uniquely to every finite extension of
211:is Henselian. That is the case if and only if
1137:has the same completion and residue field as
98:if Hensel's lemma holds. This means that if
1418:
948:
929:
832:
813:
64:
1459:Henselsche Ringe und algebraische Geometrie
1457:Kurke, H.; Pfister, G.; Roczen, M. (1975),
1290:is Henselian if and only if the associated
48:, but most authors now restrict them to be
1587:
1542:
1497:
1376:
1629:
1572:"On the theory of Henselian rings. III"
1463:VEB Deutscher Verlag der Wissenschaften
1400:
1355:
60:
37:
1666:
1603:
1566:
1527:"On the theory of Henselian rings. II"
1521:
1476:
1300:is Henselian (this is the quotient of
1094:
587:{\displaystyle v(\alpha ')=v(\alpha )}
110:, then any factorization of its image
56:
778:Henselian rings in algebraic geometry
1428:Publications Mathématiques de l'IHÉS
1077:there is a universal Henselian ring
1273:over a Henselian ring is Henselian.
1125:is an algebraic substitute for the
13:
1482:"On the theory of Henselian rings"
1225:. It is not "universal" as it has
14:
1685:
1283:of a Henselian ring is Henselian.
1245:local rings, such as the ring of
140:A Henselian local ring is called
1361:"On maximally central algebras."
1068:
806:is a Henselian local ring, and
207:is said to be Henselian if its
36:holds. They were introduced by
1337:
1042:
992:
986:
954:{\displaystyle \{U_{i}\to X\}}
942:
906:
876:
870:
838:{\displaystyle \{U_{i}\to X\}}
826:
685:
666:{\displaystyle v\circ \sigma }
581:
575:
566:
555:
424:
412:
369:
357:
70:
1:
1330:
344:
7:
1604:Nagata, Masayoshi (1975) ,
1531:Nagoya Mathematical Journal
1486:Nagoya Mathematical Journal
1408:Encyclopedia of Mathematics
1365:Nagoya Mathematical Journal
1319:then it is Henselian since
1306:ideal of nilpotent elements
1232:
845:is a Nisnevich covering of
773:are known to be conjugated.
10:
1690:
1343:A. J. Engler, A. Prestel,
1051:{\displaystyle U_{i}\to X}
915:{\displaystyle U_{i}\to X}
1632:Anneaux locaux henséliens
1544:10.1017/s002776300001802x
1499:10.1017/s0027763000015439
1378:10.1017/s0027763000010114
998:{\displaystyle X=Spec(R)}
882:{\displaystyle X=Spec(R)}
437:is a Henselian field and
1630:Raynaud, Michel (1970),
1401:Danilov, V. I. (2001) ,
498:{\displaystyle \alpha '}
1420:Grothendieck, Alexandre
1133:. The Henselization of
1121:. The Henselization of
1109:. The Henselization of
640:{\displaystyle K^{alg}}
607:{\displaystyle \sigma }
518:{\displaystyle \alpha }
450:{\displaystyle \alpha }
330:{\displaystyle K^{sep}}
297:{\displaystyle K^{alg}}
40:, who named them after
1589:10.1215/kjm/1250776700
1161:, normal, regular, or
1052:
1019:
999:
955:
916:
883:
839:
800:
767:
747:
727:
699:
698:{\displaystyle v|_{K}}
667:
641:
608:
588:
539:
519:
499:
471:
451:
431:
396:
376:
331:
298:
265:
245:
225:
201:
178:
1269:A local ring that is
1053:
1020:
1000:
956:
917:
884:
840:
801:
786:in the sense that if
768:
748:
728:
700:
668:
642:
609:
589:
540:
520:
500:
472:
452:
432:
430:{\displaystyle (K,v)}
397:
377:
375:{\displaystyle (K,v)}
332:
299:
266:
246:
226:
202:
179:
1192:strict Henselization
1029:
1009:
965:
926:
893:
849:
810:
790:
757:
737:
733:, the extensions of
709:
677:
651:
618:
598:
549:
529:
509:
484:
461:
441:
409:
386:
354:
308:
275:
255:
235:
215:
191:
168:
157:abuse of terminology
122:) into a product of
1674:Commutative algebra
1177:formal power series
1167:ring of polynomials
1073:For any local ring
726:{\displaystyle L/K}
673:is an extension of
1640:10.1007/BFb0069571
1440:10.1007/BF02732123
1221:of order prime to
1099:local homomorphism
1048:
1015:
995:
951:
912:
889:, then one of the
879:
835:
796:
784:Nisnevich topology
763:
743:
723:
695:
663:
637:
604:
584:
535:
515:
495:
467:
457:is algebraic over
447:
427:
392:
372:
327:
294:
261:
241:
221:
197:
174:
142:strictly Henselian
1649:978-3-540-05283-8
1615:978-0-88275-228-0
1568:Nagata, Masayoshi
1523:Nagata, Masayoshi
1478:Nagata, Masayoshi
1018:{\displaystyle R}
799:{\displaystyle R}
766:{\displaystyle L}
746:{\displaystyle v}
538:{\displaystyle K}
477:, then for every
470:{\displaystyle K}
395:{\displaystyle K}
264:{\displaystyle K}
244:{\displaystyle K}
224:{\displaystyle v}
200:{\displaystyle v}
177:{\displaystyle K}
65:Grothendieck 1967
59:, Chapter VII), (
1681:
1660:
1626:
1600:
1591:
1563:
1546:
1518:
1501:
1473:
1453:
1452:
1451:
1442:, archived from
1415:
1397:
1380:
1348:
1341:
1097:, such that any
1093:, introduced by
1057:
1055:
1054:
1049:
1041:
1040:
1024:
1022:
1021:
1016:
1005:of a local ring
1004:
1002:
1001:
996:
961:of the spectrum
960:
958:
957:
952:
941:
940:
921:
919:
918:
913:
905:
904:
888:
886:
885:
880:
844:
842:
841:
836:
825:
824:
805:
803:
802:
797:
772:
770:
769:
764:
752:
750:
749:
744:
732:
730:
729:
724:
719:
704:
702:
701:
696:
694:
693:
688:
672:
670:
669:
664:
646:
644:
643:
638:
636:
635:
613:
611:
610:
605:
593:
591:
590:
585:
565:
544:
542:
541:
536:
524:
522:
521:
516:
504:
502:
501:
496:
494:
476:
474:
473:
468:
456:
454:
453:
448:
436:
434:
433:
428:
401:
399:
398:
393:
381:
379:
378:
373:
336:
334:
333:
328:
326:
325:
303:
301:
300:
295:
293:
292:
270:
268:
267:
262:
250:
248:
247:
242:
230:
228:
227:
222:
206:
204:
203:
198:
183:
181:
180:
175:
150:separably closed
104:monic polynomial
75:In this article
1689:
1688:
1684:
1683:
1682:
1680:
1679:
1678:
1664:
1663:
1650:
1616:
1449:
1447:
1352:
1351:
1342:
1338:
1333:
1325:
1299:
1261:complex numbers
1235:
1229:automorphisms.
1071:
1036:
1032:
1030:
1027:
1026:
1010:
1007:
1006:
966:
963:
962:
936:
932:
927:
924:
923:
900:
896:
894:
891:
890:
850:
847:
846:
820:
816:
811:
808:
807:
791:
788:
787:
780:
758:
755:
754:
738:
735:
734:
715:
710:
707:
706:
689:
684:
683:
678:
675:
674:
652:
649:
648:
625:
621:
619:
616:
615:
599:
596:
595:
558:
550:
547:
546:
530:
527:
526:
510:
507:
506:
487:
485:
482:
481:
462:
459:
458:
442:
439:
438:
410:
407:
406:
387:
384:
383:
355:
352:
351:
347:
315:
311:
309:
306:
305:
282:
278:
276:
273:
272:
256:
253:
252:
236:
233:
232:
216:
213:
212:
192:
189:
188:
169:
166:
165:
137:of local rings.
73:
67:, Chapter 18).
46:non-commutative
12:
11:
5:
1687:
1677:
1676:
1662:
1661:
1648:
1627:
1614:
1601:
1564:
1519:
1474:
1454:
1416:
1398:
1350:
1349:
1335:
1334:
1332:
1329:
1328:
1327:
1323:
1309:
1297:
1284:
1277:
1274:
1267:
1264:
1263:are Henselian.
1253:
1250:-adic integers
1239:
1234:
1231:
1219:roots of unity
1070:
1067:
1063:Ă©tale topology
1047:
1044:
1039:
1035:
1014:
994:
991:
988:
985:
982:
979:
976:
973:
970:
950:
947:
944:
939:
935:
931:
911:
908:
903:
899:
878:
875:
872:
869:
866:
863:
860:
857:
854:
834:
831:
828:
823:
819:
815:
795:
779:
776:
775:
774:
762:
742:
722:
718:
714:
692:
687:
682:
662:
659:
656:
634:
631:
628:
624:
603:
583:
580:
577:
574:
571:
568:
564:
561:
557:
554:
534:
514:
493:
490:
466:
446:
426:
423:
420:
417:
414:
403:
391:
371:
368:
365:
362:
359:
346:
343:
342:
341:
338:
324:
321:
318:
314:
291:
288:
285:
281:
260:
240:
220:
209:valuation ring
196:
173:
153:
138:
131:
72:
69:
38:Azumaya (1951)
34:Hensel's lemma
22:Henselian ring
9:
6:
4:
3:
2:
1686:
1675:
1672:
1671:
1669:
1659:
1655:
1651:
1645:
1641:
1637:
1633:
1628:
1625:
1621:
1617:
1611:
1607:
1602:
1599:
1595:
1590:
1585:
1581:
1577:
1573:
1569:
1565:
1562:
1558:
1554:
1550:
1545:
1540:
1536:
1532:
1528:
1524:
1520:
1517:
1513:
1509:
1505:
1500:
1495:
1491:
1487:
1483:
1479:
1475:
1472:
1468:
1464:
1460:
1455:
1446:on 2016-03-03
1445:
1441:
1437:
1433:
1429:
1425:
1421:
1417:
1414:
1410:
1409:
1404:
1403:"Hensel ring"
1399:
1396:
1392:
1388:
1384:
1379:
1374:
1370:
1366:
1362:
1358:
1357:Azumaya, GorĂ´
1354:
1353:
1346:
1345:Valued fields
1340:
1336:
1322:
1318:
1315:has only one
1314:
1310:
1307:
1303:
1296:
1293:
1289:
1285:
1282:
1278:
1275:
1272:
1268:
1265:
1262:
1258:
1254:
1251:
1249:
1244:
1240:
1237:
1236:
1230:
1228:
1224:
1220:
1216:
1215:-adic numbers
1214:
1209:
1208:automorphisms
1205:
1201:
1197:
1193:
1190:, called the
1189:
1184:
1182:
1178:
1174:
1171:
1168:
1164:
1160:
1156:
1152:
1148:
1144:
1140:
1136:
1132:
1128:
1124:
1120:
1116:
1112:
1108:
1104:
1100:
1096:
1095:Nagata (1953)
1092:
1088:
1087:Henselization
1085:, called the
1084:
1081:generated by
1080:
1076:
1069:Henselization
1066:
1064:
1059:
1045:
1037:
1033:
1025:, one of the
1012:
989:
983:
980:
977:
974:
971:
968:
945:
937:
933:
909:
901:
897:
873:
867:
864:
861:
858:
855:
852:
829:
821:
817:
793:
785:
760:
740:
720:
716:
712:
690:
680:
660:
657:
654:
632:
629:
626:
622:
601:
578:
572:
569:
562:
559:
552:
532:
512:
491:
488:
480:
464:
444:
421:
418:
415:
404:
389:
366:
363:
360:
349:
348:
339:
322:
319:
316:
312:
289:
286:
283:
279:
258:
238:
218:
210:
194:
187:
171:
164:
163:
158:
154:
151:
147:
146:residue field
143:
139:
136:
132:
129:
125:
121:
117:
113:
109:
105:
101:
97:
93:
90:
89:maximal ideal
86:
83:A local ring
82:
81:
80:
78:
68:
66:
62:
58:
53:
51:
47:
43:
39:
35:
31:
27:
23:
19:
1631:
1605:
1579:
1575:
1534:
1530:
1489:
1485:
1458:
1448:, retrieved
1444:the original
1431:
1427:
1406:
1368:
1364:
1344:
1339:
1320:
1312:
1301:
1294:
1292:reduced ring
1287:
1247:
1222:
1212:
1203:
1199:
1195:
1191:
1187:
1185:
1181:power series
1179:(the formal
1169:
1150:
1146:
1138:
1134:
1130:
1122:
1110:
1106:
1102:
1090:
1086:
1082:
1078:
1074:
1072:
1060:
781:
350:Assume that
160:
141:
127:
119:
115:
111:
107:
99:
95:
91:
84:
74:
61:Raynaud 1970
54:
25:
21:
15:
1606:Local rings
1371:: 119–150,
1326:is a field.
1317:prime ideal
1227:non-trivial
1143:flat module
1119:isomorphism
304:, resp. to
271:, resp. to
71:Definitions
57:Nagata 1975
50:commutative
42:Kurt Hensel
26:Hensel ring
18:mathematics
1582:: 93–101,
1450:2007-12-09
1331:References
1200:non-unique
1155:Noetherian
1127:completion
1113:is unique
345:Properties
94:is called
30:local ring
1553:0027-7630
1508:0027-7630
1492:: 45–57,
1434:: 5–361,
1413:EMS Press
1387:0027-7630
1243:Hausdorff
1241:Complete
1173:localized
1163:excellent
1141:and is a
1043:→
943:→
907:→
827:→
661:σ
658:∘
602:σ
579:α
560:α
513:α
489:α
479:conjugate
445:α
186:valuation
96:Henselian
32:in which
1668:Category
1570:(1959),
1537:: 1–19,
1525:(1954),
1480:(1953),
1422:(1967),
1359:(1951),
1281:quotient
1271:integral
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77:rings
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