Singularity categories via the derived quotient

Abstract

Given a noncommutative partial resolution $A={\mathrm{End}}_R(R\oplus M)$ of a Gorenstein singularity $R$, we show that the relative singularity category ${\Delta }_R(A)$ of Kalck-Yang is controlled by a certain connective dga $A{/}^{\mathbb{L}}AeA$, the derived quotient of Braun-Chuang-Lazarev. We think of $A{/}^{\mathbb{L}}AeA$ as a kind of `derived exceptional locus' of the partial resolution $A$, as we show that it can be thought of as the universal dga fitting into a suitable recollement. This theoretical result has geometric consequences. When $R$ is an isolated hypersurface singularity, it follows that the singularity category $D_{\mathrm{sg}}(R)$ is determined completely by $A{/}^{\mathbb{L}}AeA$, even when $A$ has infinite global dimension. Thus our derived contraction algebra classifies threefold flops, even those $X\to \mathrm{Spec}(R)$ where $X$ has only terminal singularities. This gives a solution to the strongest form of the derived Donovan-Wemyss conjecture, which we further show is the best possible classification result in this singular setting.

46 pages. This is an improvement of the second half of arXiv:1810.10060, split off for publication. v2: proof of 6.4.6 improved, other minor refinements. To appear in Advances in Mathematics. v3: error in published version corrected, see http://mattbooth.info/papers/singcats-errata.pdf for the full errata

References in corpus (5)

• Noncommutative Knörrer type equivalences via noncommutative resolutions of singularities
• Relative singularity categories III: Cluster resolutions
• The derived contraction algebra
• The derived deformation theory of a point
• Homological epimorphisms and homotopy epimorphisms

Cited by in corpus (3)

• The derived deformation theory of a point
• The Donovan--Wemyss Conjecture via the Derived Auslander--Iyama Correspondence
• Singularity categories via higher McKay quivers with potential