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- W4300991611 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." @default.
- W4300991611 created "2022-10-04" @default.
- W4300991611 creator A5035832068 @default.
- W4300991611 date "2017-09-14" @default.
- W4300991611 modified "2023-10-14" @default.
- W4300991611 title "Relative Singularity Categories" @default.
- W4300991611 doi "https://doi.org/10.48550/arxiv.1709.04753" @default.
- W4300991611 hasPublicationYear "2017" @default.
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