“School of Mathematics”
Back to Papers HomeBack to Papers of School of Mathematics
| Paper IPM / M / 18027 |
|
| Abstract: | |
|
Let $\Lambda$ be an artin algebra and $\mathcal{M}$ be an $n$-cluster tilting subcategory of $\Lambda$-mod with $n \geq 2$. From the viewpoint of higher homological algebra, a question that naturally arose in [17] is when $\mathcal{M}$ induces an $n$-cluster tilting subcategory of $\Lambda$-Mod. In this paper, we answer this question and explore its connection to Iyamaâ??s question on the finiteness of $n$-cluster tilting subcategories of $\Lambda$-mod. In fact, our theorem reformulates Iyamaâ??s question in terms of the vanishing of Ext; and highlights its relation with the rigidity of filtered colimits of $\mathcal{M}$. Also, we show that ${\rm Add}(\mathcal{M})$ is an $n$-cluster tilting subcategory of $\Lambda$-Mod if and only if ${\rm Add}(\mathcal{M})$ is a maximal $n$-rigid subcategory of $\Lambda$-Mod if and only if $\mathcal{M}$ is of finite type if and only if ${\rm Ext}_{\Lambda}^1({\underrightarrow{\lim}}\mathcal{M}, {\underrightarrow{\lim}}\mathcal{M})=0$. Moreover, we present several equivalent conditions for Iyamaâ??s question which shows the relation of Iyama's question with different subjects in representation theory such as purity and covering theory.
Download TeX format |
|
| back to top | |
