The rigidity of filtered colimits of n-cluster tilting subcategories

Let Lambda be an artin algebra and M be an n-cluster tilting subcategory of Lambda-mod with n >= 2. From the viewpoint of higher homological algebra, a question that naturally arose in [17] is when M induces an n-cluster tilting subcategory of Lambda-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 Lambda-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 M. Also, we show that Add(M) is an n-cluster tilting subcategory of Lambda-Mod if and only if Add(M) is a maximal n-rigid subcategory of Lambda-Mod if and only if {X is an element of Lambda-Mod | Ext(Lambda)(i)(M,X)=0 for all 0<i<n}subset of Add(M) if and only if M is of finite type if and only if Ext(Lambda)(1)(lim(->)M,lim(->)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.