The rigidity of filtered colimits of n -cluster tilting subcategories

Abstract

Let Λ be an artin algebra and be an n-cluster tilting subcategory of Λ-mod with. From the viewpoint of higher homological algebra, a question that naturally arose in Ebrahimi and Nasr-Isfahani (The completion of d-abelian categories. J. Algebra 645 (2024), 143-163) is when induces an n-cluster tilting subcategory of Λ-Mod. In this article, 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. Also, we show that is an n-cluster tilting subcategory of Λ-Mod if and only if is a maximal n-rigid subcategory of Λ-Mod if and only if if and only if is of finite type if and only if. 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. © The Author(s), 2025.