Given a noncommutative partial resolution $A={\mathrm{End}}_R(R\oplus M)$ of a Gorenstein singularity R, we show that the relative singularity category ${\mathrm{\Delta }}_R(A)$ 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.

给出非交换部分分辨率 $\mathrm{一个}={\mathrm{结束}}_{\mathrm{[R}}（\mathrm{[R}\oplus \mathrm{中号}）$Gorenstein奇点R的方程，我们证明了相对奇点类别${\mathrm{\Delta }}_{\mathrm{[R}}（\mathrm{一个}）$ Kalck–Yang的控制由某个结缔组织dga控制 $\mathrm{一个}{/}^{\mathbb{大号}}\mathrm{一个}Ë\mathrm{一个}$，即Braun–Chuang–Lazarev的导出商。我们想到$\mathrm{一个}{/}^{\mathbb{大号}}\mathrm{一个}Ë\mathrm{一个}$作为部分分辨率A的一种“派生例外轨迹” ，正如我们所展示的，它可以被认为是适合于适当折衷的通用dga。该理论结果具有几何后果。当R是一个孤立的超曲面奇点时，奇点类别$d_{\mathrm{g}}（\mathrm{[R}）$ 完全取决于 $\mathrm{一个}{/}^{\mathbb{大号}}\mathrm{一个}Ë\mathrm{一个}$，即使A具有无限的全局维。因此，我们得出的收缩代数将三级触发器分类，即使是那些$X\to \mathrm{规格}（\mathrm{[R}）$其中X仅具有末端奇异点。这为派生的Donovan-Wemyss猜想的最强形式提供了一种解决方案，我们进一步证明了在这种奇异设置下最好的分类结果。