Marina Godinho (University of Glasgow) – A twist on ring morphisms

In this talk, I will show that a ring morphism $p:A ⟶ B$ satisfying certain mild assumptions induces a derived endomorphism of $A$ and a derived endomorphism of $B$, which are closely related. In fact, the derived endomorphism of $A$ is the twist around the restriction of scalars functor, and the derived endomorphism of $B$ is the corresponding cotwist. These endomorphisms are autoequivalences in certain settings, one of which is that of Frobenius exact categories. More precisely, assume that $A$ is the endomorphism algebra of an object in a Frobenius exact category satisfying mild assumptions and B is the corresponding stable endomorphism algebra. Then, if $B$ is "$n$-relatively spherical", I will show that both the twist and cotwist are equivalences. In fact, when $B$ is finite dimensional, "3-relatively spherical" is equivalent to self-injective, and the cotwist turns out to be a shift of the Nakayama autoequivalence of $B$. This technology can be used to construct new derived autoequivalences for very singular varieties.