330:) occurs in it. This invariant is additive on short exact sequences of finitely generated torsion modules (though it is not additive on short exact sequences of finitely generated modules). It vanishes if and only if the finitely generated torsion module is finitely generated as a module over the subring
267:
Many of the modules over this algebra that occur in
Iwasawa theory are finitely generated torsion modules. The structure of such modules can be described as follows. A quasi-isomorphism of modules is a homomorphism whose kernel and cokernel are both finite groups, in other words modules with support
264:] occurs in it. This is well-defined and is additive for short exact sequences of finitely-generated modules. The rank of a finitely generated module is zero if and only if the module is a torsion module, which happens if and only if the support has dimension at most 1.
503:
890:
653:-group. These are the finitely generated modules whose support has dimension at most 0. Such modules are Artinian and have a well defined length, which is finite and additive on short exact sequences.
649:
If the rank, the μ-invariant, and the λ-invariant of a finitely generated module all vanish, the module is finite (and conversely); in other words its underlying abelian group is a finite abelian
633:
349:
751:
1008:
559:
936:
776:
pointed out that the
Iwasawa's result could be deduced from standard results about the structure of modules over integrally closed Noetherian rings such as the Iwasawa algebra.
1043:
965:
1169:
1213:
268:
either empty or the height 2 prime ideal. For any finitely generated torsion module there is a quasi-isomorphism to a finite sum of modules of the form
1292:
1257:
814:
306:, that is uniquely defined up to multiplication by a unit. The ideal generated by the characteristic power series is called the
205:
797:
343:
is the sum of the degrees of the distinguished polynomials that occur. In other words, if the module is pseudo-isomorphic to
310:
of the
Iwasawa module. More generally, any generator of the characteristic ideal is called a characteristic power series.
570:
498:{\displaystyle \bigoplus _{i}\mathbf {Z} _{p}\!]/(p^{\mu _{i}})\oplus \bigoplus _{j}\mathbf {Z} _{p}\!]/(f_{j}^{m_{j}})}
198:
298:) occurs in the module is well defined and independent of the composition series. The torsion module therefore has a
698:
224:
970:
792:
dividing the order of the ideal class group of the cyclotomic field generated by the roots of unity of order
521:
1316:
906:
188:
1140:
1122:
159:
1130:
1087:
1019:
941:
194:
1241:
1282:
1253:
1222:
1188:
1148:
1271:
1178:
1105:
1097:
91:
1300:
1267:
1234:
1200:
1160:
1078:
Coates, John; Schneider, Peter; Sujatha, Ramdorai (2003), "Modules over
Iwasawa algebras",
1296:
1275:
1263:
1249:
1230:
1196:
1156:
1109:
191:
135:
83:
25:
1144:
1248:, Grundlehren der Mathematischen Wissenschaften, vol. 323 (1st ed.), Berlin:
108:
1286:
1310:
1208:
638:
In terms of the characteristic power series, the μ-invariant is the minimum of the (
55:
40:
1183:
667:
for the element 1+γ+γ+...+γ where γ is a topological generator of Γ. Iwasawa (
285:
is a generator of a height 1 prime ideal. Moreover, the number of times any module
209:
227:(polynomials with leading coefficient 1 and all other coefficients divisible by
112:
1101:
212:
over a complete local ring that the prime ideals of this ring are as follows:
32:
1226:
1192:
1152:
642:-adic) valuations of the coefficients and the λ-invariant is the power of
317:
of a finitely-generated torsion module is the number of times the module
80:
1285:(1958), "Classes des corps cyclotomiques (d'après K. Iwasawa) Exp.174",
1167:
Iwasawa, Kenkichi (1959), "On Γ-extensions of algebraic number fields",
1135:
1092:
675:
is a finitely generated torsion module over the
Iwasawa algebra and
302:, a formal power series given by the product of the power series
255:
of a finitely generated module is the number of times the module
1045:-modules are classified up to so-called pseudo-null modules.
885:{\displaystyle \Lambda (G):=\varprojlim _{H}\mathbf {Z} _{p}}
1244:; Schmidt, Alexander; Wingberg, Kay (2000), "Chapter 5",
1123:"Ring-theoretic properties of Iwasawa algebras: a survey"
179:. The isomorphism is given by identifying 1 +
1022:
973:
944:
909:
817:
701:
573:
524:
352:
141:is isomorphic to the additive group of the ring of
628:{\displaystyle \lambda =\sum _{j}m_{j}\deg(f_{j}).}
1037:
1002:
959:
930:
884:
745:
627:
553:
497:
111:, and non-commutative Iwasawa algebras of compact
90:. Commutative Iwasawa algebras were introduced by
458:
448:
388:
378:
804:Higher rank and non-commutative Iwasawa algebras
903:-adic Lie group. The case above corresponds to
808:More general Iwasawa algebras are of the form
772:. Iwasawa's original argument was ad hoc, and
1170:Bulletin of the American Mathematical Society
967:up to pseudo-isomorphism is possible in case
8:
779:In particular this applies to the case when
746:{\displaystyle e_{n}=\mu p^{n}+\lambda n+c}
223:), and the ideals generated by irreducible
1182:
1134:
1091:
1021:
991:
986:
981:
972:
943:
922:
917:
908:
871:
859:
854:
844:
834:
816:
722:
706:
700:
613:
594:
584:
572:
545:
535:
523:
484:
479:
474:
462:
439:
434:
427:
409:
404:
392:
369:
364:
357:
351:
1054:
1003:{\displaystyle G=\mathbf {Z} _{p}^{n}.}
668:
95:
46:coefficients that take the topology of
554:{\displaystyle \mu =\sum _{i}\mu _{i}}
119:
773:
7:
1214:Publications Mathématiques de l'IHÉS
760:sufficiently large, where μ, λ, and
646:at which that minimum first occurs.
515:are distinguished polynomials, then
938:. A classification of modules over
158:) is isomorphic to the ring of the
1121:Ardakov, K.; Brown, K. A. (2006),
1023:
945:
931:{\displaystyle G=\mathbf {Z} _{p}}
818:
14:
982:
918:
855:
435:
365:
183:with a topological generator of
50:into account. More precisely, Λ(
1209:"Groupes analytiques p-adiques"
1184:10.1090/S0002-9904-1959-10317-7
206:Weierstrass preparation theorem
187:. This ring is a 2-dimensional
1293:Société Mathématique de France
1032:
1026:
954:
948:
879:
865:
827:
821:
800:states that μ=0 in this case.
619:
606:
492:
467:
459:
455:
449:
445:
417:
397:
389:
385:
379:
375:
1:
234:Height 2: the maximal ideal (
134:In the special case when the
1246:Cohomology of Number Fields
1038:{\displaystyle \Lambda (G)}
960:{\displaystyle \Lambda (G)}
300:characteristic power series
199:unique factorization domain
1333:
1288:Séminaire Bourbaki, Vol. 5
1068:, Theorems 4, 5, §VII.4.4.
1062:Bourbaki, Nicolas (1972),
798:Ferrero–Washington theorem
247:Finitely generated modules
1102:10.1017/S1474748003000045
225:distinguished polynomials
216:Height 0: the zero ideal.
788:is the largest power of
154:, the Iwasawa algebra Λ(
79: runs through the
1207:Lazard, Michel (1965),
656:
170:] in one variable over
126:Iwasawa algebra of the
1080:J. Inst. Math. Jussieu
1039:
1004:
961:
932:
886:
747:
629:
555:
499:
197:, and in particular a
31:is a variation of the
1127:Documenta Mathematica
1040:
1005:
962:
933:
887:
748:
630:
556:
500:
219:Height 1: the ideal (
116:-adic analytic groups
1020:
1012:For non-commutative
971:
942:
907:
815:
699:
571:
522:
350:
308:characteristic ideal
204:It follows from the
16:In mathematics, the
1145:2005math.....11345A
1064:Commutative Algebra
996:
491:
160:formal power series
118:were introduced by
58:of the group rings
1295:, pp. 83–93,
1283:Serre, Jean-Pierre
1035:
1000:
980:
957:
928:
882:
849:
842:
743:
625:
589:
551:
540:
495:
470:
432:
362:
195:regular local ring
98:) in his study of
1259:978-3-540-66671-4
835:
833:
671:) showed that if
657:Iwasawa's theorem
580:
531:
423:
353:
1324:
1303:
1278:
1242:Neukirch, Jürgen
1237:
1203:
1186:
1163:
1138:
1113:
1112:
1095:
1075:
1069:
1067:
1066:, Paris: Hermann
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478:
466:
444:
443:
438:
431:
416:
415:
414:
413:
396:
374:
373:
368:
361:
84:normal subgroups
1332:
1331:
1327:
1326:
1325:
1323:
1322:
1321:
1307:
1306:
1281:
1260:
1250:Springer-Verlag
1240:
1221:(26): 389–603,
1206:
1166:
1120:
1117:
1116:
1077:
1076:
1072:
1061:
1060:
1056:
1051:
1018:
1017:
969:
968:
940:
939:
916:
905:
904:
853:
813:
812:
806:
787:
764:depend only on
718:
702:
697:
696:
684:
666:
659:
609:
590:
569:
568:
541:
520:
519:
513:
480:
433:
405:
400:
363:
348:
347:
338:
325:
293:
276:
263:
249:
178:
169:
153:
145:-adic integers
136:profinite group
132:
106:
66:
26:profinite group
18:Iwasawa algebra
12:
11:
5:
1330:
1328:
1320:
1319:
1309:
1308:
1305:
1304:
1279:
1258:
1238:
1204:
1177:(4): 183–226,
1164:
1115:
1114:
1070:
1053:
1052:
1050:
1047:
1034:
1031:
1028:
1025:
999:
994:
989:
984:
979:
976:
956:
953:
950:
947:
925:
920:
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912:
893:
892:
881:
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829:
826:
823:
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805:
802:
783:
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753:
742:
739:
736:
733:
730:
725:
721:
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714:
709:
705:
680:
662:
658:
655:
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624:
621:
616:
612:
608:
605:
602:
597:
593:
587:
583:
579:
576:
562:
561:
548:
544:
538:
534:
530:
527:
511:
506:
505:
494:
487:
483:
477:
473:
469:
465:
461:
457:
454:
451:
447:
442:
437:
430:
426:
422:
419:
412:
408:
403:
399:
395:
391:
387:
384:
381:
377:
372:
367:
360:
356:
334:
321:
289:
272:
259:
248:
245:
244:
243:
232:
217:
174:
165:
149:
131:
130:-adic integers
124:
109:Iwasawa theory
107:extensions in
102:
62:
13:
10:
9:
6:
4:
3:
2:
1329:
1318:
1317:Number theory
1315:
1314:
1312:
1302:
1298:
1294:
1290:
1289:
1284:
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1273:
1269:
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1261:
1255:
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1243:
1239:
1236:
1232:
1228:
1224:
1220:
1216:
1215:
1210:
1205:
1202:
1198:
1194:
1190:
1185:
1180:
1176:
1172:
1171:
1165:
1162:
1158:
1154:
1150:
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1137:
1132:
1128:
1124:
1119:
1118:
1111:
1107:
1103:
1099:
1094:
1089:
1086:(1): 73–108,
1085:
1081:
1074:
1071:
1065:
1058:
1055:
1048:
1046:
1029:
1015:
1010:
997:
992:
987:
977:
974:
951:
923:
913:
910:
902:
899:is a compact
898:
876:
872:
868:
860:
850:
845:
839:
836:
830:
824:
811:
810:
809:
803:
801:
799:
795:
791:
786:
782:
777:
775:
771:
767:
763:
759:
740:
737:
734:
731:
728:
723:
719:
715:
712:
707:
703:
695:
694:
693:
691:
687:
683:
678:
674:
670:
665:
654:
652:
647:
645:
641:
622:
614:
610:
603:
600:
595:
591:
585:
581:
577:
574:
567:
566:
565:
546:
542:
536:
532:
528:
525:
518:
517:
516:
514:
485:
481:
475:
471:
463:
452:
440:
428:
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420:
410:
406:
401:
393:
382:
370:
358:
354:
346:
345:
344:
342:
337:
333:
329:
324:
320:
316:
311:
309:
305:
301:
297:
292:
288:
284:
280:
275:
271:
265:
262:
258:
254:
246:
241:
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211:
207:
202:
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196:
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186:
182:
177:
173:
168:
164:
161:
157:
152:
148:
144:
140:
137:
129:
125:
123:
121:
120:Lazard (1965)
117:
115:
110:
105:
101:
97:
93:
89:
85:
82:
78:
74:
70:
65:
61:
57:
56:inverse limit
53:
49:
45:
43:
38:
34:
30:
27:
23:
19:
1287:
1245:
1218:
1212:
1174:
1168:
1136:math/0511345
1126:
1093:math/0110342
1083:
1079:
1073:
1063:
1057:
1013:
1011:
900:
896:
894:
807:
793:
789:
784:
780:
778:
774:Serre (1958)
769:
765:
761:
757:
755:
689:
685:
681:
676:
672:
663:
660:
650:
648:
643:
639:
637:
563:
509:
507:
340:
335:
331:
327:
322:
318:
314:
312:
307:
303:
299:
295:
290:
286:
282:
278:
273:
269:
266:
260:
256:
252:
250:
239:
235:
228:
220:
210:power series
203:
184:
180:
175:
171:
166:
162:
155:
150:
146:
142:
138:
133:
127:
113:
103:
99:
87:
76:
72:
68:
63:
59:
51:
47:
41:
36:
28:
21:
17:
15:
768:and not on
341:λ-invariant
315:μ-invariant
208:for formal
1276:0948.11001
1110:1061.11060
1049:References
688:has order
508:where the
192:Noetherian
33:group ring
1291:, Paris:
1227:1618-1913
1193:0002-9904
1153:1431-0635
1024:Λ
946:Λ
851:
840:←
819:Λ
732:λ
716:μ
604:
582:∑
575:λ
543:μ
533:∑
526:μ
425:⨁
421:⊕
407:μ
355:⨁
54:) is the
1311:Category
1129:: 7–33,
281:) where
189:complete
1301:1603459
1268:1737196
1235:0209286
1201:0124316
1161:2290583
1141:Bibcode
661:Write ν
94: (
92:Iwasawa
24:) of a
1299:
1274:
1266:
1256:
1233:
1225:
1199:
1191:
1159:
1151:
1108:
895:where
796:. The
339:. The
1131:arXiv
1088:arXiv
692:then
75:) as
44:-adic
39:with
1254:ISBN
1223:ISSN
1189:ISSN
1149:ISSN
756:for
669:1959
564:and
313:The
253:rank
251:The
96:1959
81:open
1272:Zbl
1179:doi
1106:Zbl
1098:doi
837:lim
601:deg
326:]/(
294:]/(
277:]/(
86:of
35:of
1313::
1297:MR
1270:,
1264:MR
1262:,
1252:,
1231:MR
1229:,
1219:26
1217:,
1211:,
1197:MR
1195:,
1187:,
1175:65
1173:,
1157:MR
1155:,
1147:,
1139:,
1125:,
1104:,
1096:,
1082:,
1016:,
831::=
679:/ν
242:).
231:).
201:.
122:.
20:Λ(
1181::
1143::
1133::
1100::
1090::
1084:2
1033:)
1030:G
1027:(
1014:G
998:.
993:n
988:p
983:Z
978:=
975:G
955:)
952:G
949:(
924:p
919:Z
914:=
911:G
901:p
897:G
880:]
877:H
873:/
869:G
866:[
861:p
856:Z
846:H
828:)
825:G
822:(
794:p
790:p
785:n
781:e
770:n
766:X
762:c
758:n
741:c
738:+
735:n
729:+
724:n
720:p
713:=
708:n
704:e
690:p
686:X
682:n
677:X
673:X
664:n
651:p
644:T
640:p
623:.
620:)
615:j
611:f
607:(
596:j
592:m
586:j
578:=
547:i
537:i
529:=
512:j
510:f
493:)
486:j
482:m
476:j
472:f
468:(
464:/
460:]
456:]
453:T
450:[
446:[
441:p
436:Z
429:j
418:)
411:i
402:p
398:(
394:/
390:]
386:]
383:T
380:[
376:[
371:p
366:Z
359:i
336:p
332:Z
328:p
323:p
319:Z
304:f
296:f
291:p
287:Z
283:f
279:f
274:p
270:Z
261:p
257:Z
240:T
238:,
236:p
229:p
221:p
185:G
181:T
176:p
172:Z
167:p
163:Z
156:G
151:p
147:Z
143:p
139:G
128:p
114:p
104:p
100:Z
88:G
77:H
73:H
71:/
69:G
67:(
64:p
60:Z
52:G
48:G
42:p
37:G
29:G
22:G
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