论文标题
增强的auslander-reiten二元性和奇异定理的奇异性类别
Enhanced Auslander-Reiten duality and Morita theorem for singularity categories
论文作者
论文摘要
我们建立了一个莫里塔定理，以在戈伦斯坦环的奇异性类别之间构建三角等效性，以及田地上有限维代数的群集类别，更强烈的是，在其规范的DG增强之间的准等效度更高。更准确地说，我们证明，只要我们发现Gorenstein环的分级DG奇异性类别与有限维代数的派生类别之间的准等效性，就可以通过找到单个倾斜对象来完成的等值。 我们的结果是基于有关集群类别和奇异类别的DG增强功能的两个关键定理，它们具有独立的兴趣。首先，我们给出一个莫里塔型定理，该定理实现了某些$ \ mathbb {z} $ - 分级的DG类别作为DG轨道类别。其次，我们表明，对称订单的奇异性类别的规范DG增强具有bimodule calabi-yau属性，该属性提高了奇异性类别上的古典奥斯兰德 - 瑞登双重性。 我们将结果应用于诸如Gorenstein尺寸的戒指等类别的戒指，最多$ 1 $，商的奇异性和Geigle-Lenzing完整的交叉点，包括有限或无限的Grassmannian群集类别，以实现其奇异性类别作为有限尺寸的群集类别。
We establish a Morita theorem to construct triangle equivalences between the singularity categories of (commutative and non-commutative) Gorenstein rings and the cluster categories of finite dimensional algebras over fields, and more strongly, quasi-equivalences between their canonical dg enhancements. More precisely, we prove that such an equivalence exists as soon as we find a quasi-equivalence between the graded dg singularity category of a Gorenstein ring and the derived category of a finite dimensional algebra which can be done by finding a single tilting object. Our result is based on two key theorems on dg enhancements of cluster categories and of singularity categories, which are of independent interest. First we give a Morita-type theorem which realizes certain $\mathbb{Z}$-graded dg categories as dg orbit categories. Secondly, we show that the canonical dg enhancements of the singularity categories of symmetric orders have the bimodule Calabi-Yau property, which lifts the classical Auslander-Reiten duality on singularity categories. We apply our results to such classes of rings as Gorenstein rings of dimension at most $1$, quotient singularities, and Geigle-Lenzing complete intersections, including finite or infinite Grassmannian cluster categories, to realize their singularity categories as cluster categories of finite dimensional algebras.
