André–Quillen cohomology explained

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 and are sometimes called Lichtenbaum–Schlessinger functors T⁰, T¹, T², and the higher groups were defined independently by and 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 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, while Quillen notates the same group as . The qth André–Quillen cohomology group is:

D ^{q(B/A,} M) = H ^{q(A,} B, M) \stackrel{ def } H^q(\operatorname_A(P, M)).

Let 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} ⊗ _{B} M).

See also

• Cotangent complex
• Deformation Theory
• Exalcomm

References

Generalizations

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