Twisted (co)homological stability for monoids of endomorphisms
type: Item
instance of: scholarly article
title: Twisted (co)homological stability for monoids of endomorphisms
zbMATH Open document ID: 0789.18008
DOI: 10.1007/BF01444912
author: Stanislaw Betley
author: Teimuraz Pirashvili
published in: Mathematische Annalen
publication date: 1994-06-06T00:00:00Z
full work available at URL: https://eudml.org/doc/165067
review text: Let \(R\) be a ring with 1 and \(M_ n(R)\) be the multiplicative monoid of \(n \times n\)-matrices over \(R\). Let \(\mathbb{Z} [M_ n(R)]\) be its monoid ring. Let \(D:(R\)-\(\bmod) ^{op} \times (R\)-\(\bmod) \to Ab\) be a functor. This paper is devoted to studying the Hochschild homology \(HH_ *(\mathbb{Z} [M_ n(R)], D(R^ n,R^ n))\), where \(M_ n(R)\) acts on \(D(R^ n,R^ n)\) from both sides by an obvious action. It is shown that these groups stabilize when \(n \to \infty\) for all rings \(R\) and almost any functor \(D\). This work was motivated by the recent development of topological Hochschild homology theory. It has turned out that the homotopy groups of \(THH(R)\) are the same as \(HH_ *(\mathbb{Z} [M_ \infty],-)\) groups with correctly chosen coefficients. Here \(M_ \infty=colim M_ n(R)\).
Mathematics Subject Classification ID: 18G15
Mathematics Subject Classification ID: 16E40
zbMATH DE Number: 537234
zbMATH Keywords: Hochschild homology
zbMATH Keywords: homotopy groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 537234