Derived equivalences arising from cotilting modules and finiteness conditions
Algebra and Geometry Seminar
6th December 2017, 2:30 pm – 3:30 pm
Howard House, 4th Floor Seminar Room
The techniques of tilting theory provide an elegant formalism for constructing derived equivalences. In turn this yields many important bridges between representation theory and other areas of mathematics. A well-known instance of such a bridge (due to Beilinson, 1978) connects finite-dimensional algebras with Grothendieck categories arising in algebraic geometry: a tilting sheaf in the category of quasi-coherent sheaves on projective n-space gives rise to a derived equivalence with the nth Beilinson algebra.
More recently, Stovicek has shown that, for an arbitrary ring R, we can detect the existence of such derived equivalences by the presence of a (possibly large) cotilting module. More precisely, there exists a Grothendieck category G with a tilting object T giving rise to an equivalence RHom(T,-) : D(G) —> D(ModR) if and only if there exists a cotilting module C in ModR.
In this talk we will discuss this perspective on tilting equivalences and go on to present joint work with Lidia Angeleri Hügel in which we explore properties of the cotilting module C that completely determine finiteness conditions on the category G.