Recollements induced by monomorphism categories. (English) Zbl 1490.18016
Summary: Let \(R\) be any associative ring, \(n\) a positive integer and \(\mathcal{X}\) a full subcategory of the category of \(R\)-modules. The monomorphism category \(\mathcal{S}_n(\mathcal{X})\) of \(\mathcal{X}\) consists of all the objects \(( X_i, \varphi_i)\), where \(X_i \in \mathcal{X}\) for each \(1 \leqslant i \leqslant n\), \(\varphi_i : X_i \to X_{i + 1}\) is a monomorphism and \(\text{Coker} \varphi_i \in \mathcal{X}\) for each \(1 \leqslant i \leqslant n - 1\), which was introduced by Zhang. We mainly prove that there exists a recollement of the stable category \(\underline{\mathcal{S}_{n + 1} ( \mathcal{X} )}\) relative to the ones \(\underline{\mathcal{S}_n ( \mathcal{X} )}\) and \(\underline{\mathcal{X}}\) for any Frobenius subcategory \(\mathcal{X}\) and any positive integer \(n\). To realize this purpose, we construct a recollement in the context of comma categories. As its application, it is shown that there are recollements induced by the monomorphism category of Gorenstein projective modules (resp., Ding projective modules, Gorenstein AC-projective modules, Gorenstein flat-cotorsion modules, and Gorenstein flat modules) and a hereditary Hovey triple in the category of \(R\)-modules, respectively.
MSC:
| 18G65 | Stable module categories |
| 18G25 | Relative homological algebra, projective classes (category-theoretic aspects) |
| 16D40 | Free, projective, and flat modules and ideals in associative algebras |
| 16E05 | Syzygies, resolutions, complexes in associative algebras |
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