On the structure theory of the Iwasawa algebra of a p-adic Lie group

By O. Venjakob. 2001

148 citations recorded by the source corpus.

Abstract.This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, Λ of a p-adic analytic group G. For G without any p-torsion element we prove that Λ is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudo-nullΛ-module. This is classical when G=ℤkp for some integer k≥1, but was previously unknown in the non-commutative case. Then the category of Λ-modules up to pseudo-isomorphisms is studied and we obtain a weak structure theorem for the ℤp-torsion part of a finitely generated Λ-module. We also prove a local duality theorem and a version of Auslander-Buchsbaum equality. The arithmetic applications to the Iwasawa theory of abelian varieties are published elsewhere.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1e99a33451e88669d1c52de0cf9f6856ffbfadf5

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In Geometry and Topology

1. R. Sujatha
2. O. Venjakob
3. John Coates
4. P. Schneider
5. Peter Scholze
6. T. Fukaya
7. M. Emerton
8. K. Ardakov

In Mathematical Physics

1. R. Sujatha
2. O. Venjakob
3. John Coates
4. P. Schneider
5. Peter Scholze
6. T. Fukaya
7. Kiran S. Kedlaya
8. M. Emerton

Citations in the corpus

• Fine Selmer groups of elliptic curves over p-adic Lie extensions (2005, background): “Is it true that the Galois group Y( F∞) of K(F∞) over F∞ is always pseudo-null as a �(G) -module (see [ 28 ] for a discussion of the notion of a pseudo-null module for Iwasawa algebras)? Surprisingly,Y.”
• Links Between Cyclotomic and GL2 Iwasawa Theory (2003, background): “Now the μ-invariant of W , which we shall denote by μG(W ), can be defined in various equivalent fashions (see [22], [11]) in terms of the structure theory of the Λ(G)-module W (p).”
• A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory (2002, background): “.10 and its proof, both A and B are Auslander regular and without zero divisor. Thus there is a dimension theory for A- and B-modules, for a thorough treatment of which we refer the reader to [10] or [28]. The previous lemma shows that the canonical (co-)dimension filtrations of M ǫ A-modR considered”
• On torsion in the cohomology of locally symmetric varieties (2013, background): “The results of [63] imply that if X is of codimension c, then E(X) = 0 for j < c, E(X) is of codimension (exactly) c, and E(X) is of codimension ≥ j for j > c.”
• The GL2 Main Conjecture for Elliptic Curves without Complex Multiplication (2004, background): “If L is a finite extension of Q in F∞ such that GL is pro-p, the μ-invariant of X viewed as a Λ(GL)-module, is defined in [19] and [36].”
• Computations in non‐commutative Iwasawa theory (2005, unknown): “There is an explicit formula for χna(E/k) in terms of χcyc(E/k) due to Hachimori and Venjakob ([26], thm 4.1 and the note following it).”