The cyclic homology of an exact category
type: Item
instance of: scholarly article
title: The cyclic homology of an exact category
zbMATH Open document ID: 0807.19002
DOI: 10.1016/0022-4049(94)90091-4
author: Randy McCarthy
published in: Journal of Pure and Applied Algebra
publication date: 1995-02-19T00:00:00Z
review text: Algebraic \(K\)-theory is at once an invariant of a ring and of an exact category, that is, the category of finitely generated modules over a ring. Constructions of homomorphisms between algebraic \(K\)-theory and other invariants of rings such as the Dennis trace map and the generalized Chern character, usually proceed through the ring-theoretic definition of \(K\)-theory. In this paper the author obtains all of the relevant additional structure on the invariants of rings that are most closely related to algebraic \(K\)-theory, that is, the Hochschild homology and the (negative, periodic) cyclic homology, by starting with an exact category. This not only generalizes the definitions of these invariants but it provides a uniform foundation for the relevant additional structure and provides means for further study of the relations between these invariants through categorical constructions. Let \(k\) denote a commutative ring with unit. The ingredients for this synthesis are the \(S\)-construction of Waldhausen applied to a category with cofibrations and the additive cyclic nerve of a small \(k\)-linear category \({\mathcal C}\); this is a cyclic \(k\)-modulo which at level \(n\) is given by \[ \text{CN}_ n ({\mathcal C}) = \bigoplus \Hom_{\mathcal C} (C_ 1,C_ 0) \otimes_ k \Hom_{\mathcal C} (C_ 2,C_ 1) \otimes_ k \cdots \otimes_ k \Hom_{\mathcal C} (C_ 0,C_ n), \] where the sum is over \(n + 1\)-tuples of objects in \({\mathcal C}\). The face maps are given by composition, the degeneracy maps by inserting an identity map, and the twist map is given by applying the cyclic permutation to the tensor product. The split Hochschild homology groups of \({\mathcal C}\) are given by the homology of the additive cyclic nerve. The composition of Waldhausen's \(S\)-construction applied to an exact category \({\mathcal E}\) followed by the additive cyclic nerve gives a simplicial cyclic abelian group, and the Hochschild homology of the exact category is given by \[ HH_ n ({\mathca…
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 18G60
zbMATH DE Number: 584667
zbMATH Keywords: Hochschild homology
zbMATH Keywords: cyclic homology
zbMATH Keywords: exact category
zbMATH Keywords: Dennis trace map
zbMATH Keywords: Chern character
reviewed by: John McCleary
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90091-4
OpenAlex ID: W2008711423
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description: scientific article; zbMATH DE number 584667