Derived Categories of Modules and Coherent Sheaves

Abstract

Abstract We present recent results on derived categories of modules and coherent sheaves, namely, tame–wild dichotomy and semi-continuity theorem for derived categories over finite dimensional algebras, as well as explicit calculations for derived categories of modules over nodal rings and of coherent sheaves over projective configurations of types A and A. This paper is a survey of some recent results on the structure of derived categories obtained by the author in collaboration with Viktor Bekkert and Igor Burban [6, 11, 12]. The origin of this research was the study of Cohen–Macaulay modules and vector bundles by Gert-Martin Greuel and myself [27, 28, 29, 30] and some ideas from the work of Huisgen-Zimmermann and Saorin [42]. Namely, I understood that the technique of “matrix problems,” briefly explained below in subsection 2.3, could be successfully applied to the calculations in derived categories, almost in the same way as it was used in the representation theory of finite-dimensional algebras, in study of Cohen–Macaulay modules, etc. The first step in this direction was the semi-continuity theorem for derived categories [26] presented in subsection 2.1. Then Bekkert and I proved the tame–wild dichotomy for derived categories over finite dimensional algebras (see subsection 2.2). At the same time, Burban and I described the indecomposable objects in the derived categories over nodal rings (see Section 3) and projective configurations of types A and A (see Section 4).