Finite generation of Hochschild homology algebras
type: Item
instance of: scholarly article
title: Finite generation of Hochschild homology algebras
zbMATH Open document ID: 0966.13009
author: Luchezar L. Avramov
author: Srikanth B. Iyengar
published in: Inventiones Mathematicae
publication date: 2001-08-05T00:00:00Z
review text: The authors, in this paper, attempt to prove the converse of the following classical result due to Hochschild, Kostant and Rosenberg: Let \(S\) be a commutative algebra over a commutative Noetherian ring \(k\). If \(S\) is smooth over \(k\), then the \(S\)-module \(\Omega^1_{S/k}\) of Kähler differentials is projective and the canonical homomorphism \(W^*_{S/k}: \wedge^*_S \Omega^1_{S/k} \to\text{HH}_*(S/k)\) of graded algebras is bijective. The authors prove the following converses of the theorem quoted above. (i) If \(S\) is a flat \(k\)-algebra essentially of finite type over \(k\) and the \(S\)-algebra HH\(_*(S/k)\) is finitely generated, then \(S\) is smooth over \(k\). (ii) If \(S\) is a flat \(k\)-algebra essentially of finite type over \(k\) and if HH\(_{2i-1} (S/k)= 0=\text{HH}_{2j}(S/k)\) for some positive integers \(i,j\), then \(S\) is smooth over \(k\).
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 13N05
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 18G15
zbMATH DE Number: 1443594
zbMATH Keywords: \(DG\)-algebra
zbMATH Keywords: Hochschild homology
zbMATH Keywords: module of Kähler differentials
zbMATH Keywords: smoothness of flat algebra
reviewed by: Veereshwar A. Hiremath
MaRDI profile type: Publication
full work available at URL: s002220000051
OpenAlex ID: W1978852029
DOI: 10.1007/S002220000051
description: scientific article; zbMATH DE number 1443594