Abstract

Smooth and proper DG-categories are noncommutative versions of smooth and proper schemes which also include finite dimensional algebras of finite global dimension. Kuznetsov and Shinder defined reflexive DG-categories as a generalisation; they include all projective schemes and all finite dimensional algebras.  Smooth and proper DG-categories can be characterised as the dualizable objects in the monoidal category of DG-categories localised at Morita equivalences. The main result I’ll talk about is a monoidal characterisation of reflexive DG-categories. This provides a conceptual explanation for why there is some common information between D^{b(mod A) and D}perf(A) for a finite dimensional algebra A and between D^{b(coh X) and D}perf(X) for a projective scheme X. One can use this approach to prove that the Hochschild cohomology of a reflexive DG-category is isomorphic to that of its derived category of cohomologically finite modules.