Hideto Asashiba (Shizuoka)

Derived equivalences and smash products.

Abstract:

Throughout this talk k is a commutative ring and G is a group. Denote by G-GrCat the 2-category of G-graded small k-categories and (weak) degree-preserving functors defined in the paper [A generalization of Gabriel's Galois covering functors II ]. In the paper [A generalization of Gabriel's Galois covering functors and derived equivalences ] (a final form in [Gluing derived equivalences together]) we investigated when the orbit categories of a pair of derived equivalent small k-categories with G-actions are derived equivalent. Here we consider the converse. By a 2-categorical Cohen-Montgomery duality proved in [A generalization of Gabriel's Galois covering functors II ], this problem is reduced to the following. Let A and B be in G-GrCat, and assume that A and B are derived equivalent. Then under which condition are the smash products A#G and B#G derived equivalent? Our solution is as follows.

Theorem.

Let A and B be as above, and assume that they are derived equivalent. If there exists a tilting subcategory P for A consisting of G-gradable complexes, and if B is equivalent in the 2-category G-GrCat to P with a G-grading defined by the canonical G-covering (Q, 1): A#G → A, then the smash products A#G and B#G are derived equivalent.