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The Lie algebra structure of tangent cohomology and deformation theory. (English) Zbl 0576.17008

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 \(\operatorname{Hom}(\Gamma A,A)\) 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 \(\Gamma A\).

MSC:

17B55 Homological methods in Lie (super)algebras
17B56 Cohomology of Lie (super)algebras
55P62 Rational homotopy theory
55S30 Massey products
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