Non-commutative Fitting invariants and annihilation of class groups
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.
scientific article; zbMATH DE number 5718179
scientific article