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. The André–Quillen cohomology groups are 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:

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

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

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

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

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

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

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, JSTOR 1994516, 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, doi:10.1017/CBO9781139644136, ISBN 978-0-521-43500-0, MR 1269324