ABSTRACT
Introduction 23 1. Backstro¨m rings 24 2. Nodal rings 25 3. Examples 29 3.1. Simple node 29 3.2. Dihedral algebra 32 3.3. Gelfand problem 33 4. Projective configurations 36 5. Configurations of type A and A˜ 37 6. Application: Cohen-Macaulay modules over surface singularities 43 References 45
Introduction
This paper is devoted to recent results on explicit calculations in derived categories of modules and coherent sheaves. The idea of this approach is actually not new and was effectively used in several questions of module theory (cf. e.g. [10, 12, 13, 7]). Nevertheless it was somewhat unexpected and successful that the same technique could be applied to derived categories, at least in the case of rings and curves with “simple singularities.” We present here two cases: nodal rings and configurations of projective lines of types A and A˜, when these calculations can be carried out up to a result, which can be presented in more or less distinct form, though it involves rather intricate combinatorics of a special sort of matrix problems, namely “bunches of semi-chains” [4] (or, equivalently, “clans” [8]). In Sections 1 and 4 we give a general construction of “categories of triples,” which are a connecting link between derived categories and matrix problems, while in Sections 2 and 5 this construction is applied to nodal rings and configurations of types A˜. Section 3 contains examples of calculations for concrete rings and Section 5 also presents those for nodal cubic. We tried to choose typical examples, which allow to better understand the general procedure of passing from combinatorial data to complexes. Section 6 contains an application to Cohen-Macaulay modules over surface singularities, which was in fact the origin of investigations of vector bundles over projective curves in [13].
