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 \(\Gamma A\).


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


[1] André, M., L’algèbre de Lie d’un anneau local, Symp. math., IV, 337-375, (1970)
[2] André, M., Méthode simpliciale en algèbre homologique et algèbre commutative, () · Zbl 0154.01402
[3] Douady, A., Obstruction primaire à la déformation, Seminaire Henri CARTAN, (1960/61), exposé 4 · Zbl 0156.42803
[4] Gerstenhaber, M., On the deformation of rings and algebras, III, Ann. math., 88, 1-34, (1968) · Zbl 0182.05902
[5] Harrison, D.K., Commutative algebra and cohomology, Trans. AMS, 104, 191-204, (1962) · Zbl 0106.25703
[6] Lichtenbaum, S.; Schlessinger, M., The cotangent complex of a morphism, Trans. AMS, 128, 41-70, (1967) · Zbl 0156.27201
[7] Michaelis, W., Lie coalgebras, Adv. in math., 38, 1-54, (1980) · Zbl 0451.16006
[8] Nijenhuis, A.; Nijenhuis, A.; Richardson, R.W., Cohomology and deformation of algebraic structures, Bull. AMS, 70, 406-411, (1964), Math. Inst., Univ van Amsterdam · Zbl 0138.26301
[9] Palamodov, V.P., Cohomology of analytic algebras, Trudy moskow math. soc., Trans. Moscow math. soc., 44, 1-61, (1983) · Zbl 0596.32012
[10] Quillen, D., On the (co-)homology of commutative rings, Proc. symp. pure math., 17, 65-87, (1970)
[11] Quillen, D., Rational homotopy theory, Ann. math., 90, 205-295, (1969) · Zbl 0191.53702
[12] M. Schlessinger and J.D. Stasheff, Deformation theory and rational homotopy type, Publ. Math. IHES, to appear. · Zbl 0576.17008
[13] Tate, J., Homology of Noetherian rings and local rings, Illinois J. math., 1, 14-27, (1957) · Zbl 0079.05501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.