Singularity categories via the derived quotient

Abstract

Given a noncommutative partial resolution A=EndR(R⊕M) of a Gorenstein singularity R, we show that the relative singularity category ΔR(A) of Kalck–Yang is controlled by a certain connective dga A/LAeA, the derived quotient of Braun–Chuang–Lazarev. We think of A/LAeA 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 Dsg(R) is determined completely by A/LAeA, even when A has infinite global dimension. Thus our derived contraction algebra classifies threefold flops, even those X→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.