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 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:

$0 \to \operatorname{Der}_B(C, M) \to \operatorname{Der}_A(C, M) \to \operatorname{Der}_A(B, M).$

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 H^q(A, B, M), while Quillen notates the same group as D^q(B/A, M). The qth André–Quillen cohomology group is:

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

The qth André–Quillen homology group is:

$D_q(B/A, M) = H_q(A, B, M) \stackrel{\text{def}}{=} H_q(\Omega_{P/A} \otimes_B M).$

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

$D^q(B/A, M) = H^q(\operatorname{Hom}_B(L_{B/A}, M)),$

$D_q(B/A, M) = H_q(L_{B/A} \otimes_B M).$

References

• André, Michel (1974), Homologie des Algèbres Commutatives, Grundlehren der mathematischen Wissenschaften 206, Springer-Verlag
• Lichtenbaum, Stephen; Schlessinger, M. (1967), "The cotangent complex of a morphism", Transactions of the American Mathematical Society 128: 41–70, doi:10.2307/1994516, ISSN 0002-9947, MR 0209339
• Quillen, Daniel G., Homology of commutative rings, unpublished notes, archived from the original on April 20, 2015
• Quillen, Daniel (1970), On the (co-)homology of commutative rings, Proc. Symp. Pure Mat. XVII, American Mathematical Society
• Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, ISBN 978-0-521-43500-0, MR 1269324