# Pure semisimple $n$-cluster tilting subcategories

## TL;DR

This paper explores pure semisimple $n$-cluster tilting subcategories within higher homological algebra, establishing conditions for their structure and generalizing classical results on pure semisimplicity of Artin algebras.

## Contribution

It introduces the concept of pure semisimple $n$-abelian categories and characterizes $n$-cluster tilting subcategories in terms of direct sums and local finiteness, extending Auslander's classical results.

## Key findings

- $ ext{Mod-}\Lambda$-category is pure semisimple iff modules are direct sums of finitely generated modules.
- $Add( ext{m})$ is $n$-cluster tilting iff $ ext{m}$ has an additive generator.
- Generalizes classical pure semisimplicity results for Artin algebras.

## Abstract

From the viewpoint of higher homological algebra, we introduce pure semisimple $n$-abelian category, which is analogs of pure semisimple abelian category. Let $\Lambda$ be an Artin algebra and $\mathcal{M}$ be an $n$-cluster tilting subcategory of $Mod$-$\Lambda$. We show that $\mathcal{M}$ is pure semisimple if and only if each module in $\mathcal{M}$ is a direct sum of finitely generated modules. Let $\mathfrak{m}$ be an $n$-cluster tilting subcategory of $mod$-$\Lambda$. We show that $Add(\mathfrak{m})$ is an $n$-cluster tilting subcategory of $Mod$-$\Lambda$ if and only if $\mathfrak{m}$ has an additive generator if and only if $Mod(\mathfrak{m})$ is locally finite. This generalizes Auslander's classical results on pure semisimplicity of Artin algebras.