The Lie algebra structure of tangent cohomology and deformation theory
type: Item
instance of: scholarly article
title: The Lie algebra structure of tangent cohomology and deformation theory
zbMATH Open document ID: 0576.17008
DOI: 10.1016/0022-4049(85)90019-2
author: Michael Schlessinger
author: Jim Stasheff
published in: Journal of Pure and Applied Algebra
publication date: 1985-01-01T00:00:00Z
review text: The main goal of the paper is to present Harrison cohomology in characteristic zero as the homology of a natural differential graded Lie algebra associated to an augmented commutative algebra \(A\). It turns out that for the algebra \(A\) Harrison and Quillen cohomologies agree for the coefficient in any \(A\)-module \(M\). This approach elucidates the differential graded Lie structure of the cohomology and substantiates the existence of Massey-Lie brackets in the cohomology. The basic point of view is that of multiplicative resolutions of \(A\), i.e. a differential graded commutative algebra \({\mathcal A}\to A\) inducing \(H(\mathcal A)\approx A\). For such resolutions, the deformations of A correspond to changes in the differential of \(\mathcal A\). The authors exhibit a particular resolution which makes the comparison with Harrison cohomology quite transparent if considering Lie coalgebras. Then the Harrison complex is precisely \(\Hom(\Gamma A,A)\) where \(\Gamma A\) is the free graded Lie coalgebra on the augmentation ideal of A considered to have degree 1. This construction gives a computation for the tangent cohomology as the cohomology of the Lie algebra of ``outer derivations'' of the construction \(\Gamma A\).
reviewed by: Alice Fialowski
Mathematics Subject Classification ID: 17B55
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 55S30
zbMATH DE Number: 3920741
zbMATH Keywords: deformation theory
zbMATH Keywords: Harrison cohomology
zbMATH Keywords: characteristic zero
zbMATH Keywords: differential graded Lie algebra
zbMATH Keywords: Massey-Lie brackets
zbMATH Keywords: multiplicative resolutions
zbMATH Keywords: free graded Lie coalgebra
zbMATH Keywords: tangent cohomology
Wikidata QID: Q115364456
MaRDI profile type: Publication
full work available at URL: 0022-4049(85)90019-2
OpenAlex ID: W2084566257
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description: scientific article; zbMATH DE number 3920741