# 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.

## Motivation

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 DerA(B, M). Before the general definitions of André and Quillen, it was known for a long time that given morphisms of commutative rings ABC and a C-module M, there is a three-term exact sequence of derivation modules:

${\displaystyle 0\to \operatorname {Der} _{B}(C,M)\to \operatorname {Der} _{A}(C,M)\to \operatorname {Der} _{A}(B,M).}$

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.

## Definition

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 Hq(A, B, M), while Quillen notates the same group as Dq(B/A, M). The qth André–Quillen cohomology group is:

${\displaystyle D^{q}(B/A,M)=H^{q}(A,B,M){\stackrel {\text{def}}{=}}H^{q}(\operatorname {Der} _{A}(P,M)).}$

The qth André–Quillen homology group is:

${\displaystyle D_{q}(B/A,M)=H_{q}(A,B,M){\stackrel {\text{def}}{=}}H_{q}(\Omega _{P/A}\otimes _{B}M).}$

Let LB/A denote the relative cotangent complex of B over A. Then we have the formulas:

${\displaystyle D^{q}(B/A,M)=H^{q}(\operatorname {Hom} _{B}(L_{B/A},M)),}$
${\displaystyle D_{q}(B/A,M)=H_{q}(L_{B/A}\otimes _{B}M).}$