André–Quillen cohomology

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 Lichtenbaum & Schlessinger (1967) and are sometimes called Lichtenbaum–Schlessinger functors T⁰, T¹, T², and the higher groups were defined independently by Michel André and by Daniel Quillen using methods of homotopy theory. It comes with a parallel homology theory called André–Quillen homology.

Let A be a commutative ring, B be an A-algebra, and M be a B-module. André–Quillen cohomology is the derived functors of the derivation functor Der_A(B, M). 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. André–Quillen cohomology extends this exact sequence even further. In the zeroth degree, it is the module of derivations; in the first degree, it is Exalcomm; and in the second degree, it is the second degree Lichtenbaum–Schlessinger functor.

Let B be an A-algebra, and let M be a B-module. 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 H^q(A, B, M), while Quillen notates the same group as D^q(B/A, M). The qth André–Quillen cohomology group is:

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