Singular Curves and Quasi-hereditary Algebras

Abstract

In this article we construct a categorical resolution of singularities of an excellent reduced curve X, introducing a certain sheaf of orders on X. This categorical resolution is shown to be a recollement of the derived category of coherent sheaves on the normalization of X and the derived category of finite length modules over a certain artinian quasi–hereditary ring Q depending purely on the local singularity types of X. Using this technique, we prove several statements on the Rouquier dimension of the derived category of coherent sheaves on X. Moreover, in the case X is rational and projective we construct a finite dimensional quasi–hereditary algebra Λ such that the triangulated category Perf(X) embeds into D(Λ −mod) as a full subcategory.