André–Quillen cohomology

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 DerA(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 Hq(A, B, M), while Quillen notates the same group as Dq(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)).

Let LB/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

Generalizations

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