The rigidity of filtered colimits of n-cluster tilting subcategories

Abstract

Let $Λ$ be an artin algebra and $\mathcal{M}$ be an n-cluster tilting subcategory of $Λ$-mod with $n\ge 2$. 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 $Λ$-Mod. In this paper, we answer this question and explore its connection to Iyama's question on the finiteness of n-cluster tilting subcategories of $Λ$-mod. In fact, our theorem reformulates Iyama'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 Add$(\mathcal{M})$ is an n-cluster tilting subcategory of $Λ$-Mod if and only if Add$(\mathcal{M})$ is a maximal n-rigid subcategory of $Λ$-Mod if and only if $\lbrace X\in Λ$-Mod$~|~ {\rm Ext}^i_Λ(\mathcal{M},X)=0 ~~~ {\rm for ~all}~ 0<i<n \rbrace \subseteq {\rm Add}(\mathcal{M})$ if and only if $\mathcal{M}$ is of finite type if and only if ${\rm Ext}_Λ^1({\underrightarrow{\lim}}\mathcal{M}, {\underrightarrow{\lim}}\mathcal{M})=0$. Moreover, we present several equivalent conditions for Iyama's question which shows the relation of Iyama's question with different subjects in representation theory such as purity and covering theory.