Let R be a commutative ring, and G an affine flat group scheme over R. We say that A is a (commutative) G-algebra if A is a G-module and is a (commutative) R-algebra, and the product A ⊗ A → A is G-linear. We say that M is a (G, A)-module (or G-equivariant A-module) if M is an A-module and is a G-module, and the A-action A ⊗ M → M is G-linear. A (G, A)-linear map simply means a G-linear A-linear map. Thus, we get an abelian category G,A with enough injectives. The main purpose of these notes is to discuss homological aspects of (G, A)-modules, from the viewpoint of commutative ring theory of R and A.

In particular, we study various (weak) Auslander–Buchweitz contexts which appear there. The theory of Cohen–Macaulay approximations over Cohen–Macaulay local rings by Auslander and Buchweitz [10] contributes greatly to the new developments in commutative ring theory [148]. On the other hand, their theory of approximations is given in rather general form as a theory of abelian categories [10, 11], and its applications are appearing in so many topics of algebras. (Weak) Auslander–Buchweitz contexts (1.1.12) are one of its formulations.

Auslander and Reiten [11] proved that, in the category of finite modules over a finite dimensional algebra over a field, Auslander–Buchweitz contexts and basic cotilting modules are in one-to-one correspondence. Miyachi [112] proved that we have an Auslander–Buchweitz context from a cotilting module in a rather general situation.