The adjoint cotranspose of modules with respect to subcategories. (English) Zbl 1494.18009
Summary: Let \(\mathcal{X}\) be a subcategory of left \(S\)-modules and \(_RU_S\) an \((R, S)\)-bimodule. As a generalization of an adjoint cotranspose, we introduce an adjoint \(\mathcal{X}\)-cotranspose of a left \(S\)-module relative to \(_RU_S\) and study its homological properties. Let \(\mathcal{V}\) be a subcategory of \(\mathcal{X}\). The relations between adjoint \(\mathcal{X}\)-cotransposes and adjoint \(\mathcal{V}\)-cotransposes are investigated under the condition that \(\mathcal{V}\) is a generator or cogenerator for \(\mathcal{X}\). Then we give some applications of these results to some categories of interest. In particular, the adjoint counterparts of Gorenstein cotransposes are established.
MSC:
| 18G25 | Relative homological algebra, projective classes (category-theoretic aspects) |
| 16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
| 16D90 | Module categories in associative algebras |
Keywords:
adjoint \(\mathcal{X}\)-cotranspose; semidualizing bimodule; Auslander class; Bass class; Gorenstein module; adjoint Gorenstein cotransposeReferences:
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| [18] | Yuxiao Wang, Guoqiang Zhao School of Science Hangzhou Dianzi University 310018 Hangzhou Zhejiang, P.R. China E-mail: 524517300@qq.com gqzhao@hdu.edu.cn |
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