Hochschild homology of complete intersections and smoothness
type: Item
instance of: scholarly article
title: Hochschild homology of complete intersections and smoothness
zbMATH Open document ID: 0821.13004
DOI: 10.1016/0022-4049(94)90070-1
published in: Journal of Pure and Applied Algebra
publication date: 1994-11-03T00:00:00Z
review text: Let \(k\) be a commutative ring, \(A\) a smooth \(k\)-algebra, \(\{a_ 1, \dots, a_ m \}\) a regular sequence in \(A\), and put \(B = A/(a_ 1, \dots, a_ m)\). Then, after necessary preliminaries about commutative differential graded algebras and the Hochschild (co)homology, the main result of this paper shows that when \(B\) is \(k\)-flat (resp. \(k\)- projective) and \(M\) is a \(B\)-module, the Hochschild homology \(H_ n (B,M)\) (resp. the Hochschild cohomology \(H^ n (B,M))\) is computed by using a certain commutative differential graded algebra obtained by tensoring the algebra \(\Omega^*_{A/k}\) of differentials with a divided power algebra. As an application, some criteria are given for the smoothness of \(B\) in terms of the Hochschild (co)homology modules.
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 13N05
Mathematics Subject Classification ID: 12G99
zbMATH DE Number: 681815
zbMATH Keywords: smooth algebra
zbMATH Keywords: regular sequence
zbMATH Keywords: Hochschild homology
zbMATH Keywords: Hochschild cohomology
zbMATH Keywords: differential graded algebra
author: Jorge A. Guccione
author: Juan J. Guccione
reviewed by: Akira Ooishi
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90070-1
OpenAlex ID: W1969481326
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description: scientific article; zbMATH DE number 681815