Pieter Belmans from Antwerpen

Abstract: |
The derived category of $P^2$ has a full exceptional collection, described by the famous Beilinson quiver and its relations. One can compute the Hochschild cohomology of $P^2$ as the Hochschild cohomology of this finite-dimensional algebra, for which many tools are available. Changing the relations in the quiver corresponds to considering derived categories of noncommutative planes. I will explain how these (and the noncommutative analogues of the quadric surface) are described using Artin-Schelter regular ($Z$-)algebras, and how one can use their classification to compute the Hochschild cohomology of all finite-dimensional algebras obtained in this way, exhibiting an interesting dimension drop. If time permits I will explain how it is expected that the fully faithful functor between the derived category of $P^2$ and its Hilbert scheme of two points relates their Hochschild cohomologies, and how this is expected to behave under deformation. |