nLab tangent complex

Contents

Idea

The tangent complex of an algebro-geometric object is meant to behave as the (sheaf of) sections of the tangent bundle.

At least in the generality of derived algebraic stacks $X$ , the tangent complex is equivalently (up to a shift in degree) to the module of sections of the infinitesimal disk bundle of $X$ (the formal completion of the diagonal $\Delta_X \colon X \to X \times X$ ) (Hennion 13, theorem 1).

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 $A$ .

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))