Friday, February 10, 14:00, 7.527
Hideto Asashiba (Shizuoka)
Derived equivalences and smash products.
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
Our solution is as follows.
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
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.