论文标题
奇异性类别中的倾斜物体和升级的突变
Tilting objects in singularity categories and levelled mutations
论文作者
论文摘要
我们显示了奇异性类别中倾斜对象的存在$ \ Mathsf {d} _ {\ MathSf {\ Mathsf {sg}}}^{\ Mathsf {gr}}}（eae）$与某些noetherian As-Noetherian As-Noetherian As-Noetherian As-groultar as-groularian As-groularian As-grounarian Asgular as-necrular As-necrular anggebras $ a $ and Idempotents $ e $ e $ e $。这给出了$ \ mathsf {d} _ {\ Mathsf {sg}}}^{\ Mathsf {gr}}（eae）$与派生类别的有限量化代数的类别之间的三角形。特别是，如果$ a $的贝林森代数为级别的koszul代数，我们会获得一个倾斜对象。 This generalises the existence of a tilting object in $\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(S^G)$, where $S$ is a Koszul AS-regular algebra and $G$ is a finite group acting on $S$, found by Iyama-Takahashi and Mori-Ueyama.我们的方法涉及使用$ \ Mathsf {d} _ {\ MathSf {sg}}}^{\ Mathsf {gr}}（eae）$中的$ \ Mathsf {d} _ {尾部，以及$ \ mathsf {d}^{\ operatorAtorName {b}}}倾斜对象上的平整突变（\ Mathsf {qgr} eae）$。
We show the existence of tilting objects in the singularity category $\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(eAe)$ associated to certain noetherian AS-regular algebras $A$ and idempotents $e$. This gives a triangle equivalence between $\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(eAe)$ and the derived category of a finite-dimensional algebra. In particular, we obtain a tilting object if the Beilinson algebra of $A$ is a levelled Koszul algebra. This generalises the existence of a tilting object in $\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(S^G)$, where $S$ is a Koszul AS-regular algebra and $G$ is a finite group acting on $S$, found by Iyama-Takahashi and Mori-Ueyama. Our method involves the use of Orlov's embedding of $\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(eAe)$ into $\mathsf{D}^{\operatorname{b}}(\mathsf{qgr} eAe)$, the bounded derived category of graded tails, and of levelled mutations on a tilting object of $\mathsf{D}^{\operatorname{b}}(\mathsf{qgr} eAe)$.
