$n$-cluster tilting subcategories from gluing systems of representation-directed algebras

Abstract

We present a new way to construct $n$-cluster tilting subcategories of abelian categories. Our method takes as input a direct system of abelian categories $\mathcal{A}_i$ with certain subcategories and, under reasonable conditions, outputs an $n$-cluster tilting subcategory of an admissible target $\mathcal{A}$ of the direct system. We apply this general method to a direct system of module categories $\text{mod}\Lambda_i$ of representation-directed algebras $\Lambda_i$ and obtain an $n$-cluster tilting subcategory $\mathcal{M}$ of a module category $\text{mod}\mathcal{C}$ of a locally bounded Krull-Schmidt category $\mathcal{C}$. In certain cases we also construct an admissible $\mathbb{Z}$-action of $\mathcal{C}$. Using a result of Darp\"o-Iyama, we obtain an $n$-cluster tilting subcategory of $\text{mod}(\mathcal{C}/\mathbb{Z})$ where $\mathcal{C}/\mathbb{Z}$ is the corresponding orbit category. We show that in this case $\text{mod}(\mathcal{C}/\mathbb{Z})$ is equivalent to the module category of a finite-dimensional algebra. In this way we construct many new families of representation-finite algebras whose module categories admit $n$-cluster tilting modules.