Summary: We shall report about equivalences of derived categories for module categories and in a parallel setting for coherent sheaves on algebraic varieties. The group of self-equivalences of these derived categories is introduced. The braid group of an affine Dynkin diagram of type \(\widetilde A_2\) maps to the group of auto-equivalences of the derived category of the group ring of the alternating group of degree 4 over any extension of the 2-adic integers containing a third root of unity. We study the action of the self-equivalence group on the group cohomology, first in general, then for our example and give some consequences.
For the entire collection see [Zbl 0969.00049].