In commutative algebra, André–Quillen cohomology is a theory of cohomology for commutative rings which is closely related to the cotangent complex.
The first three cohomology groups were introduced by Stephen Lichtenbaum and Michael Schlessinger (1967) and are sometimes called Lichtenbaum–Schlessinger functors T0, T1, T2, and the higher groups were defined independently by Michel André (1974) and Daniel Quillen (1970) using methods of homotopy theory.
Before the general definitions of André and Quillen, it was known for a long time that given morphisms of commutative rings A → B → C and a C-module M, there is a three-term exact sequence of derivation modules: This term can be extended to a six-term exact sequence using the functor Exalcomm of extensions of commutative algebras and a nine-term exact sequence using the Lichtenbaum–Schlessinger functors.
Let P be a simplicial cofibrant A-algebra resolution of B. André notates the qth cohomology group of B over A with coefficients in M by Hq(A, B, M), while Quillen notates the same group as Dq(B/A, M).
The qth André–Quillen cohomology group is: Let LB/A denote the relative cotangent complex of B over A.