In commutative algebra, AndrĂ©âQuillen 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 Stephen Lichtenbaum and Michael Schlessinger ( 1967) and are sometimes called LichtenbaumâSchlessinger functors T0, T1, T2, and the higher groups were defined independently by Michel AndrĂ© ( 1974) and Daniel Quillen ( 1970) using methods of homotopy theory. It comes with a parallel homology theory called AndrĂ©âQuillen homology.
Let A be a commutative ring, B be an A-algebra, and M be a B-module. The AndrĂ©âQuillen cohomology groups are the derived functors of the derivation functor DerA(B, M). Before the general definitions of AndrĂ© 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:
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 LichtenbaumâSchlessinger functors. AndrĂ©âQuillen 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 LichtenbaumâSchlessinger functor.
Let B be an A-algebra, and let M be a B-module. Let P be a simplicial cofibrant A-algebra resolution of B. AndrĂ© 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 AndrĂ©âQuillen cohomology group is:
Let LB/A denote the relative cotangent complex of B over A. Then we have the formulas:
In commutative algebra, AndrĂ©âQuillen 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 Stephen Lichtenbaum and Michael Schlessinger ( 1967) and are sometimes called LichtenbaumâSchlessinger functors T0, T1, T2, and the higher groups were defined independently by Michel AndrĂ© ( 1974) and Daniel Quillen ( 1970) using methods of homotopy theory. It comes with a parallel homology theory called AndrĂ©âQuillen homology.
Let A be a commutative ring, B be an A-algebra, and M be a B-module. The AndrĂ©âQuillen cohomology groups are the derived functors of the derivation functor DerA(B, M). Before the general definitions of AndrĂ© 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:
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 LichtenbaumâSchlessinger functors. AndrĂ©âQuillen 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 LichtenbaumâSchlessinger functor.
Let B be an A-algebra, and let M be a B-module. Let P be a simplicial cofibrant A-algebra resolution of B. AndrĂ© 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 AndrĂ©âQuillen cohomology group is:
Let LB/A denote the relative cotangent complex of B over A. Then we have the formulas: