André–Quillen cohomology

Template:Short description 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 Template:Harvs and are sometimes called Lichtenbaum–Schlessinger functors T⁰, T¹, T², and the higher groups were defined independently by Template:Harvs and Template:Harvs 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 Template:Nowrap and a C-module M, there is a three-term exact sequence of derivation modules:

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

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 Template:Nowrap, while Quillen notates the same group as Template:Nowrap. The qth André–Quillen cohomology group is:

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

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

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

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

See also

• Cotangent complex
• Deformation Theory
• Exalcomm

References

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Generalizations

• André–Quillen cohomology of commutative S-algebras
• Homology and Cohomology of E-infinity ring spectra