Non-commutative Fitting invariants and annihilation of class groups
type: Item
instance of: scholarly article
title: Non-commutative Fitting invariants and annihilation of class groups
zbMATH Open document ID: 1222.11132
author: Andreas Nickel
published in: Journal of Algebra
publication date: 2010-06-08T00:00:00Z
review text: The main goal of the paper is to introduce a generalization of the concept of Fitting ideal in the non-commutative setting. Let \(A\) be a separable (not necessarily commutative) algebra over a field \(K\) which is a quotient field of a complete local ring \(R,\) let \(\Lambda\) be a finitely generated \(R\)-order in \(A.\) For a finitely presented left \(\Lambda\)-module \(M\) the author defines the Fitting invariants of \(M\) as certain equivalence classes of submodules of the center of \(A\). Some basic properties of the Fitting invariants are proven, in particular the question of the independence of the Fitting invariants of the presentation of \(M\) is investigated. An application of the newly defined Fitting invariants is also given. If \(L/K\) is a Galois CM-extension of number fields with Galois group \(G\), then assuming the validity of the Equivariant Tamagawa Number Conjecture the author constructs certain explicit elements of the center of the algebra \(\mathbb{Z}_p[G]\) which annihilate the \(p\)-part of the ideal class group of \(L\). This construction generalizes a construction of \textit{C. Greither} [Compos. Math. 143, No. 6, 1399--1426 (2007; Zbl 1135.11059)] which was carried out in the abelian case.
reviewed by: Alexey Zykin
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 16H10
zbMATH DE Number: 5718179
zbMATH Keywords: Fitting invariant
zbMATH Keywords: equivariant \(L\)-values
zbMATH Keywords: class groups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.02.039
OpenAlex ID: W1978276013
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DOI: 10.1016/J.JALGEBRA.2010.02.039
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description: scientific article; zbMATH DE number 5718179