Title: Relative Singularity Categories,
Abstract: We study the following generalization of singularity categories. Let X be a
quasi-projective Gorenstein scheme with isolated singularities and A a
non-commutative resolution of singularities of X in the sense of Van den Bergh.
We introduce the relative singularity category as the Verdier quotient of the
bounded derived category of coherent sheaves on A modulo the category of
perfect complexes on X. We view it as a measure for the difference between X
and A. The main results of this thesis are the following.
(i) We prove an analogue of Orlov's localization result in our setup. If X
has isolated singularities, then this reduces the study of the relative
singularity categories to the affine case.
(ii) We prove Hom-finiteness and idempotent completeness of the relative
singularity categories in the complete local situation and determine its
Grothendieck group.
(iii) We give a complete and explicit description of the relative singularity
categories when X has only nodal singularities and the resolution is given by a
sheaf of Auslander algebras.
(iv) We study relations between relative singularity categories and classical
singularity categories. For a simple hypersurface singularity and its Auslander
resolution, we show that these categories determine each other.
(v) The developed technique leads to the following `purely commutative'
application: a description of Iyama & Wemyss triangulated category for rational
surface singularities in terms of the singularity category of the rational
double point resolution.
(vi) We give a description of singularity categories of gentle algebras.