Singular Curves and Quasi-hereditary Algebras

Abstract

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 b (Λ − mod) as a full subcategory.

The full exceptional collections of categorical resolutions of curves

“…Remark 4. In a recent paper [1] a result which is related to the above claim has been proved. Actually it has been proved that if X is a reduced rational curve, then there exists a categorical resolution (T , π * , π * ) of X such that T c has a tilting object, which in general does not come from an exceptional collection.…”

“…. A n } of objects such that (1) for all i one has Hom C (A i , A i ) = k and Hom C (A i , A i [l]) = 0 for all l = 0;…”

“…Actually it has been proved that if X is a reduced rational curve, then there exists a categorical resolution (T , π * , π * ) of X such that T c has a tilting object, which in general does not come from an exceptional collection. See [1] Theorem 7.4.…”

Morita theory for non-commutative noetherian schemes

“…Non-hereditary reduced non-commutative projective curves naturally arise as categorical resolutions of singularities of usual singular reduced commutative curves; see [7]. From the point of view of representation theory of finite dimensional -algebras, the so-called tame non-commutative projective nodal curves seem to be of particular importance; see [5,6].…”

“…7.6 that any two hereditary orders of the same type in the simple algebra Λ x are centrally Morita equivalent. Theorem 7.8 implies the following result, which was proven for the first time by Spieß; see[34, Proposition 2.9].…”