论文标题
由$ n $ -calabi-yau Triples产生的三角和商类别的属性
Properties of triangulated and quotient categories arising from $n$-Calabi-Yau triples
论文作者
论文摘要
在过去的20年的过程中，Fomin和Zelevinsky对群集代数的定义已通过多种方式进行了分类和概括，从而引起了群集理论。这项研究导致Iyama和Yang的广义集群类别$ \ Mathcal {t}/\ Mathcal {t}^{fd} $来自$ n $ -n $ -calabi-calabi-yau Triples $（\ Mathcal {t}在本文中，我们使用一些同源代数的经典工具对此类类别有更深入的了解$ \ MATHCAL {T}/\ MATHCAL {T}^{fd} $。 让$ k $为一个字段，$ n \ geq 3 $ a Integer和$ \ Mathcal {t} $ a $ k $ -linear三角形类别，带有三角形子类别$ \ Mathcal {t}^{t}^{fd} $ $（\ MATHCAL {T}，\ MATHCAL {T}^{fd}，\ Mathcal {M}）$是$ n $ -calabi-yau三重。在本文中，我们证明了三角形类别的某些属性$ \ MATHCAL {T} $和$ \ MATHCAL {T}/\ MATHCAL {T}^{fd} $。我们的第一个结果使用限制和colimits在这些类别中的HOM空间之间存在关系。我们的第二个结果是$ \ MATHCAL {T} $中的差距定理，显示截断三角形分裂时。 此外，我们将两个定理应用于Guo的结果提供替代证明，最初在更具体的DG $ K $ -K $ -Algebras $ a $的设置中表示，DG $ a $ mmodules的派生类别的子类别。这证明了$ \ Mathcal {t}/\ Mathcal {t}^{fd} $是hom-finite，$（n-1）$ - calabi-yau，其对象$ m $是$（n-1）$ - 集群倾斜度 - 内态倾斜和$ m $ m $ m $ \ mathcalcal calcal calcal calcal calcal calcal calcal calcal {t} t} t} $ \ MATHCAL {T}/\ MATHCAL {t}^{fd} $是同构。请注意，这些属性使$ \ Mathcal {t}/\ Mathcal {t}^{fd} $ cluster类别的概括。
The original definition of cluster algebras by Fomin and Zelevinsky has been categorified and generalised in several ways over the course of the past 20 years, giving rise to cluster theory. This study lead to Iyama and Yang's generalised cluster categories $\mathcal{T}/\mathcal{T}^{fd}$ coming from $n$-Calabi-Yau triples $(\mathcal{T}, \mathcal{T}^{fd}, \mathcal{M})$. In this paper, we use some classic tools of homological algebra to give a deeper understanding of such categories $\mathcal{T}/\mathcal{T}^{fd}$. Let $k$ be a field, $n\geq 3$ an integer and $\mathcal{T}$ a $k$-linear triangulated category with a triangulated subcategory $\mathcal{T}^{fd}$ and a subcategory $\mathcal{M}=\text{add}(M)$ such that $(\mathcal{T}, \mathcal{T}^{fd}, \mathcal{M})$ is an $n$-Calabi-Yau triple. In this paper, we prove some properties of the triangulated categories $\mathcal{T}$ and $\mathcal{T}/\mathcal{T}^{fd}$. Our first result gives a relation between the Hom-spaces in these categories, using limits and colimits. Our second result is a Gap Theorem in $\mathcal{T}$, showing when the truncation triangles split. Moreover, we apply our two theorems to present an alternative proof to a result by Guo, originally stated in a more specific setup of dg $k$-algebras $A$ and subcategories of the derived category of dg $A$-modules. This proves that $\mathcal{T}/\mathcal{T}^{fd}$ is Hom-finite and $(n-1)$-Calabi-Yau, its object $M$ is $(n-1)$-cluster tilting and the endomorphism algebras of $M$ over $\mathcal{T}$ and over $\mathcal{T}/\mathcal{T}^{fd}$ are isomorphic. Note that these properties make $\mathcal{T}/\mathcal{T}^{fd}$ a generalisation of the cluster category.
