论文标题
功能有限的心脏，负面群集类别中的简单脑海和非交叉分区
Functorially finite hearts, simple-minded systems in negative cluster categories, and noncrossing partitions
论文作者
论文摘要
令$ Q $为无环箭弹，$ w \ geq 1 $是整数。令$ \ mathsf {c} _ { - w}（\ mathbf {k} q）$为$（ - w）$ - $ \ mathbf {k} q $的cluster类别。我们表明，在$ \ mathsf {d}^b（\ Mathbf {k} q）中，简单的集合之间存在双重饲料，位于$ \ Mathsf {c} _ {c} _ { - w}的基本域中（\ Mathbf {K} Q）$。如果$ q $是dynkin，这概括了Iyama-jin的相同结果。我们证明的关键步骤是观察到，在HOM-FINITE，KRULL-SCHMIDT，$ \ MATHBF {k} $ - 线性饱和的三角形类别$ \ mathsf {d} $ in Mathsf $ nathsf $ nath if if if if if if if if if if if if if if if if if，有足够的注射剂和足够的投影。然后，我们在$ \ mathsf {c} _ { - w}（\ Mathbf {k} q）$和正$ W $ -Nononcrossing分区的$ \ mathsf {c} _ { - w}中建立了两者之间的两者。
Let $Q$ be an acyclic quiver and $w \geq 1$ be an integer. Let $\mathsf{C}_{-w} (\mathbf{k} Q)$ be the $(-w)$-cluster category of $\mathbf{k} Q$. We show that there is a bijection between simple-minded collections in $\mathsf{D}^b (\mathbf{k} Q)$ lying in a fundamental domain of $\mathsf{C}_{-w} (\mathbf{k} Q)$ and $w$-simple-minded systems in $\mathsf{C}_{-w} (\mathbf{k} Q)$. This generalises the same result of Iyama-Jin in the case that $Q$ is Dynkin. A key step in our proof is the observation that the heart $\mathsf{H}$ of a bounded t-structure in a Hom-finite, Krull-Schmidt, $\mathbf{k}$-linear saturated triangulated category $\mathsf{D}$ is functorially finite in $\mathsf{D}$ if and only if $\mathsf{H}$ has enough injectives and enough projectives. We then establish a bijection between $w$-simple-minded systems in $\mathsf{C}_{-w} (\mathbf{k} Q)$ and positive $w$-noncrossing partitions of the corresponding Weyl group $W_Q$.
