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.

中文翻译：

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纯半简单 n 簇倾斜子类别

从高等同调代数的角度，我们引入纯半单$n$-阿贝尔范畴，它是纯半单阿贝尔范畴的类比。令 $\Lambda$ 是 Artin 代数，$\mathcal{M}$ 是 $Mod$-$\Lambda$ 的 $n$-簇倾斜子类别。我们证明 $\mathcal{M}$ 是纯半简单的当且仅当 $\mathcal{M}$ 中的每个模块都是有限生成模块的直接和。令 $\mathfrak{m}$ 是 $mod$-$\Lambda$ 的 $n$-cluster 倾斜子类别。我们证明 $Add(\mathfrak{m})$ 是 $Mod$-$\Lambda$ 的 $n$-cluster 倾斜子范畴当且仅当 $\mathfrak{m}$ 具有加性生成器当且仅当$Mod(\mathfrak{m})$ 是局部有限的。这概括了 Auslander 关于 Artin 代数的纯半简单性的经典结果。