Speakers:  Yuri Drozd (Harvard University and Institute of Mathematics, Kyiv, Ukraine)

Title:  "Categorical resolution and tilting for non-commutative algebraic curves"

Abstract:  Several sorts of categorical resolutions for singular algebraic
varieties have been constructed by Kuznetsov, Lunts, van den Bergh and
other authors. Mostly, these constructions are rathre complicated, as
they use gluing of some categories, and it is not so easy to study
their structure. We propose a more “concrete” construction for the
case of singular curve, where the resulting category arises as a
category of sheaves over a non-commutative curve.

A non-commutative curve is a pair (X,A), where X is an algebraic curve
and A is a sheaf of OX-algebras coherent as a sheaf of modules. For
such a curve we construct a categorical resolution for the derived
category, calculate its global homological dimension, find a
semi-orthogonal decomposition of the derived category and obtain an
upper bound for the Rouquier dimension. If X is rational, we construct
a tilting complex that establishes a derived equivalence of the curve
(X,A) to a finite dimensional quasi-hereditary algebra.

It is a joint work with I. Burban and V. Gavran.