nLab
tangent complex

Idea

For $A$ an ordinary associative algebra, its tangent complex is its module of derivations.

For $A$ a dg-algebra, its tangent complex is the essentially the value of the derived functor of the derivations-assigning functor on $A$. This is closely related to the automorphism ∞-Lie algebra of $A$.

Related concepts

• deformation theory

• tangent complex, André-Quillen cohomology?, Hochschild cohomology

• cotangent complex, André-Quillen homology?, Hochschild homology

References

The concept goes back to

• M. Schlessinger, Jim Stasheff, The Lie algebra structure of tangent cohomology and deformation theory , J. Pure Appl. Algebra, 38(1985), 313–322.

The tangent complex of an algebra over an operad in chain complexes is discussed in section 8 of

• Vladimir Hinich, Homological algebra of homotopy algebras Communications in algebra, 25(10). 3291-3323 (1997)(arXiv:q-alg/9702015, Erratum (arXiv:math/0309453))

See also

• Jonathan Block, A. Lazarev, André-Quillen cohomology and rational homotopy of function spaces , Advances in mathematics, 193 (2005) (pdf)