Hideto Asashiba (Shizuoka)

Cohen-Montgomery duality for bimodules and applications to Morita equivalences and stable equivalences of Morita type.

Abstract:

We fix a commutative ring k and a group G. To include infinite coverings of k-algebras into consideration we usually regard k-algebras as locally bounded k-categories with finite objects, so we will work with small k-categories. For small k-categories R and S with G-actions we define G-invariant S-R-bimodules and their category denoted by S-{Mod^G}-R, and denote by R/G the orbit category of R by G. For small G-graded k-categories A and B we define G-graded B-A-bimodules and their category denoted by B-{Mod_G}-A, and denote by A#G the smash product of A and G. We then define functors (-)/G : S-{Mod^G}-R --> (S/G)-{Mod_G}-(R/G) and (-)#G : A-{Mod_G}-B --> (A#G)-{Mod^G}-(B#G), and show that they are equivalences and quasi-inverses to each other having good properties with tensor products and preserving projectivity of bimodules. We apply this to Morita equivalences and to stable equivalences of Morita type (and if possible to standard derived equivalences) to have the following type of theorems: There exists a "G-invariant stable equivalence of Morita type" between R and S if and only if there exists a "G-graded stable equivalence of Morita type" between R/G and S/G. There exists a "G-graded stable equivalence of Morita type" between A and B if and only if there exists a "G-invariant stable equivalence of Morita type" between A#G and B#G.