Note on Nakayama’s lemma for compact $\Lambda$-modules

Proceedings of the National Academy of Sciences of the United States of America, 1997

The purpose of this article is to describe certain results and conjectures concerning the structure of Galois cohomology groups and Selmer groups, especially for abelian varieties. These results are analogues of a classical theorem of Iwasawa. We formulate a very general version of the Weak Leopoldt Conjecture. One consequence of this conjecture is the nonexistence of proper ⌳-submodules of finite index in a certain Galois cohomology group. Under certain hypotheses, one can prove the nonexistence of proper ⌳-submodules of finite index in Selmer groups. An example shows that some hypotheses are needed. The results that I will describe here are motivated by a well-known theorem of Iwasawa. Let K be a finite extension of ‫.ޑ‬ Let K ϱ ͞K be the cyclotomic ‫ޚ‬ p-extension of K, where p is any prime. Thus K ϱ ʕ K(pϱ) and ⌫ ϭ Gal(K ϱ ͞K) Х ‫ޚ‬ p , the additive group of p-adic integers. We let ⌳ ϭ ‫ޚ‬ p [[⌫]] be the completed group algebra of ⌫ over ‫ޚ‬ p , which is isomorphic (noncanonically) to the formal power series ring ‫ޚ‬ p [[T]]. Let M ϱ denote the maximal abelian prop extension of K ϱ unramified outside ⌺ ϭ {p, ϱ}. Let L ϱ denote the maximal abelian prop extension of K ϱ unramified at all primes of K ϱ. Let X ϭ Gal(M ϱ ͞K ϱ) and Y ϭ Gal(L ϱ ͞K ϱ). In ref. 1, Iwasawa proves the following important result. THEOREM (Iwasawa): (i) X and Y are finitely generated ⌳-modules. (ii) Rank ⌳ (X) ϭ r 2 , where r 2 denotes the number of complex primes of K. (iii) Y is a torsion ⌳-module. (iv) X has no nonzero finite ⌳-submodules. We remark also that if K ϱ ͞K is an arbitrary ‫ޚ‬ p-extension, (i) and (iii) are true (due to Iwasawa). Statement (ii) should conjecturally be true. It is often referred to as the ''Weak Leopoldt Conjecture'' for K ϱ ͞K and has the following interpretation. Let K n denote the unique subfield of K such that K n ͞K is cyclic of degree p n. Let K n denote the compositum of all ‫ޚ‬ p-extensions of K n. Then it is known that Gal͑K n ͞K n ͒ Х ‫ޚ‬ p r 2 p n ϩ1ϩ␦n , where ␦ n Ն 0. Leopoldt's Conjecture states that ␦ n ϭ 0. The Weak Leopoldt Conjecture states that ␦ n is bounded as n 3 ϱ, which is equivalent to the assertion that rank ⌳ (X) ϭ r 2. Also if statement (ii) holds, then so does statement (iv). (See proposition 4 of ref. 2.) Returning to the cyclotomic ‫ޚ‬ p-extension K ϱ ͞K, we can restate Iwasawa's theorem in terms of the Pontryagin duals Hom͑X, ‫ޑ‬ p ‫ޚ͞‬ p ͒, Hom͑Y, ‫ޑ‬ p ‫ޚ͞‬ p ͒, which are subgroups of H 1 (G K ϱ , ‫ޑ‬ p ‫ޚ͞‬ p) ϭ Hom(Gal(K ϱ ab ͞ K ϱ), ‫ޑ‬ p ‫ޚ͞‬ p) defined by imposing certain local conditions. They are examples of what have come to be called ''Selmer

2002

The goal of this paper is to prove that the Pontryagin dual of the Selmer group over the trivializing extension of an elliptic curve without complex multiplication does not have any nonzero pseudo-null submodule. The main point is to extend the definition of pseudo-null to modules over the completed group ring Z p [[G]] of an arbitrary p-adic Lie group G without p-torsion. For this purpose we prove that Z p [[G]] is an Auslander regular ring. For the proof we also extend some results of Jannsen's homotopy theory of modules and study intensively higher Iwasawa adjoints.

Comptes Rendus Mathematique, 2016

We present several finiteness results for absolutely almost simple algebraic groups over finitely generated fields that are more general than global fields. We also discuss the relations between the various finiteness properties involved in these results, such as the properness of the global-to-local map in the Galois cohomology of a given K-group G relative to a certain natural set V of discrete valuations of K, and the finiteness of the number of isomorphism classes of K-forms of G having, on the one hand, smooth reduction at V and, on the hand, the same isomorphism classes of maximal K-tori as G. Résumé. Nous présentons plusieurs résultats de finitude pour les groupes algébriques absolument presque simples définis sur des corps de type fini plus généraux que les corps globaux. Nous discutons aussi des liens entre les propriétés de finitude divers qui entrent dans le cadre de notre analyse, tels que la propreté de l'application globale-locale dans la cohomologie galoisienne d'un K-groupe G par rapportà un ensemble convenable V de valuations discrètes de K, et la finitude du nombre de K-formes de G ayant, d'une part, bonne réduction en V , et, d'autre part, possédant les même classes d'isomorphisme de K-tores maximaux que G. même ensemble marcheégalement pour les groupes de type G 2. Dans les sections 2 et 3, il s'agit de l'analyse des groupes algébriques absolument presque simples possédant les mêmes classes d'isomorphisme de tores maximaux sur le corps de définition. Plus précisément, si G est un groupe algébrique simplement connexe absolument presque simple défini sur 1

Notices of the American Mathematical Society, 2021

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000