论文标题
通过派生商的奇异性类别
Singularity categories via the derived quotient
论文作者
论文摘要
给定戈伦斯坦奇异性$ r $ r $的非共同部分分辨率$ a = \ mathrm {end} _r（r \ oplus m）$，我们表明，相对奇异性类别$Δ_r（a）$Δ_r（a）$ a $ kalck -yang由某个连接的dga $ a/^$ a/^$ a/^kalcy a/^kalcy. Braun-Chuang Lazarev的派生商。我们认为$ a/^{\ mathbb {l}} \ kern -2pt aea $是部分分辨率$ a $的一种“派生的特殊基因座”，因为我们证明它可以将其视为通用DGA拟合适用于合适的回忆。这种理论结果具有几何后果。当$ r $是一种孤立的高表情奇异性时，奇异性类别$ d_ \ mathrm {sg}（r）$完全由$ a/^{\ mathbb {l}} \ kern -kern -2pt aea $，即使$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $。因此，我们派生的收缩代数分类了三倍的拖鞋，即使是$ x \ to \ mathrm {spec}（r）$，其中$ x $只有终端奇异性。这为派生的Donovan-Wemyss猜想的最强形式提供了解决方案，我们进一步证明这是在这种单数环境中最好的分类结果。
Given a noncommutative partial resolution $A=\mathrm{End}_R(R\oplus M)$ of a Gorenstein singularity $R$, we show that the relative singularity category $Δ_R(A)$ of Kalck-Yang is controlled by a certain connective dga $A/^{\mathbb{L}}\kern -2pt AeA$, the derived quotient of Braun-Chuang-Lazarev. We think of $A/^{\mathbb{L}}\kern -2pt AeA$ 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 $A/^{\mathbb{L}}\kern -2pt AeA$, 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.
