2017/09/14 by Martin Kalck, Kalck, Martin
Mathematics · Physics and Astronomy · #13C14 #14B05 #14E15 #14F05 #14J17 #16E65 #16G20 #18E30 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Category Theory (math.CT) #Commutative Algebra (math.AC) #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1709.04753
openalex publication_date 2017/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.