Cyclic homology of Dedekind domains
type: Item
instance of: scholarly article
title: Cyclic homology of Dedekind domains
zbMATH Open document ID: 0781.13008
DOI: 10.1007/BF00966115
published in: \(K\)-Theory
publication date: 1993-04-01T00:00:00Z
review text: In this paper the cyclic homology of \(A:=R[t]/P(t)\) is computed for an integral domain \(R\) and a monic irreducible polynomial \(P\). This is then applied to the ring of algebraic integers in an algebraic number field. In section 1 a local global principle is derived for the Hochschild homology of the integral closure \(S\) of a Dedekind domain \(R\) in a separable extension field \(L\) of the field of fractions \(K\) of \(R\). In case of perfect residue fields the homology groups are computed. Moreover, for \(a\in A\) dividing \(P'(t)\) the ring structure of the Hochschild homology with coefficients in \(A/a\cdot A\) is determined. -- Next the authors give a complex, the ranks of whose modules grow only linearly, and the homology of which is the cyclic homology of \(A\). For a Dedekind domain \(R\) with global field of fractions \(K\) and \(S\) the integral closure in a separable extension \(L\) of \(K\) the computation of the cyclic homology is reduced to that of certain cyclic algebras. The results are then applied to give a formula for cyclic homology of tamely ramified extensions and for cyclotomic extensions, by employing a graded structure. Also the cyclic homology of integral group rings of cyclic \(p\)-groups is computed. [Reviewer's remark: The results here for cyclic \(p\)-groups should generalize to blocks with normal cyclic defect groups, and may be -- via derived equivalence (or Green correspondence) -- to arbitrary blocks with cyclic defect].
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 13D25
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 11R04
zbMATH DE Number: 149775
zbMATH Keywords: Dedekind domains
zbMATH Keywords: cyclic homology
zbMATH Keywords: integral domain
zbMATH Keywords: ring of algebraic integers
zbMATH Keywords: Hochschild homology
author: Michael Larsen
author: Ayelet Lindenstrauss
reviewed by: Klaus W. Roggenkamp
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00966115
OpenAlex ID: W1989558647
description: scientific article; zbMATH DE number 149775