Pure semisimple \(n\)-cluster tilting subcategories (Q2292846)

For a fixed positive integer \(n\), \(n\)-cluster tilting subcategories of abelian categories were studied by \textit{O. Iyama} [Adv. Math. 210, No. 1, 22--50 (2007; Zbl 1115.16005); Adv. Math. 210, No. 1, 51--82 (2007; Zbl 1115.16006)]. The concept of \(n\)-abelian categories was introduced by \textit{G. Jasso} [Math. Z. 283, No. 3--4, 703--759 (2016; Zbl 1356.18005)]. In this paper, the authors define purity for (compactly generated) \(n\)-abelian categories, and call such a category pure semisimple if each of its objects is pure-projective, which is an analog of pure semisimple abelian categories. Let \(\Lambda\) be an Artin algebra, and Mod-\(\Lambda\) (resp., mod-\(\Lambda\)) the category of left (resp., finitely generated) \(\Lambda\)-modules. For a class \(\mathcal X\) of modules in Mod-\(\Lambda\), Add(\(\mathcal X\)) (resp., add(\(\mathcal X\))) will denote the full subcategory of Mod-\(\Lambda\) consisting of all direct summands of (resp., finite) direct sums of modules in \(\mathcal X\). It is shown that if \(\mathcal M\) is an \(n\)-cluster tilting subcategory of Mod-\(\Lambda\), then \(\mathcal M\) is pure semisimple if and only if each module in \(\mathcal M\) is a direct sum of finitely generated modules. If \(\mathbf m\) is an \(n\)-cluster tilting subcategory of mod-\(\Lambda\), then (\(\Lambda\), \(\mathbf m\)) is called an \(n\)-homological pair. An \(n\)-homological pair is defined to be pure semisimple if Add(\(\mathbf m\)) is an \(n\)-cluster tilting subcategory of Mod-\(\Lambda\). An \(n\)-homological pair (\(\Lambda\), \(\mathbf m\)) is said to be of finite type if \(\mathbf m\) has an additive generator. The main result of the paper states that an \(n\)-homological pair (\(\Lambda\), \(\mathbf m\)) is pure semisimple if and only if (\(\Lambda\), \(\mathbf m\)) is of finite type. The proof of the ``if'' part is based on the result that if \(M\) is a finitely generated left \(\Lambda\)-module, then Add(\(M\)) is an \(n\)-cluster tilting subcategory of Mod-\(\Lambda\) if and only if add(\(M\)) is an \(n\)-cluster tilting subcategory of mod-\(\Lambda\). The proof of the other direction, that the pure semisimplicity of (\(\Lambda\), \(\mathbf m\)) implies finite type, is based on functor category techniques as developed in [\textit{M. Auslander}, Commun. Algebra 1, 269--310 (1974; Zbl 0285.16029)].