# Knörrer periodicity and equivalences of singularity categories

*Abstract:*

The singularity category is a homological invariant of a scheme that captures the singular information. The singularity category of the type A_n Kleinian surface singularity is equivalent to that of the finite dimensional algebra k[x]/(x^n), and this can be shown via Knörrer periodicity. I will discuss joint work with Martin Kalck that extends a similar phenomenon outside of the Kleinian case to produce an equivalence between the singularity category of a cyclic quotient surface singularity and a certain finite dimensional algebra. This equivalence is constructed geometrically by considering subcategories of the derived category of a surface containing a type A configuration of rational curves , and intriguingly the finite? dimensional algebras produced are noncommutative in general.