The rigidity of filtered colimits of n-cluster tilting subcategories
Ziba Fazelpour, Alireza Nasr-Isfahani
TL;DR
This work investigates when an $n$-cluster tilting subcategory $M$ of $ leftarrow$Lambda-mod extends to an $n$-cluster tilting subcategory in $ leftarrow$Lambda-Mod and its relation to Iyama's finiteness question. It establishes that Add$(M)$ is $n$-cluster tilting in $ leftarrow$Lambda-Mod precisely when $M$ is of finite type, and it characterizes this via Ext vanishing and the structure of the filtered colimit $ leftarrow o M$. The paper proves that finite type is equivalent to $ leftarrow o M$ being $n$-cluster tilting (and hence $n$-rigid) in $ leftarrow$Lambda-Mod with ${ m Ext}^1( leftarrow o M, leftarrow o M)=0$, and it shows that every pure submodule of Add$(M)$-modules splits under these conditions. It also connects these algebraic finiteness questions to purity, Mittag-Leffler conditions, and covering theory, offering several equivalent formulations of Iyama's question and illuminating the interplay between filtered colimits and higher homological properties.
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.