On a natural duality between Grothendieck categories. (English) Zbl 1153.16005

Castaño-Iglesias, F.

Commun. Algebra 36, No. 6, 2079-2091 (2008).

R. R. Colby and K. R. Fuller [J. Algebra 242, No. 1, 146-159 (2001; Zbl 0990.16009)] characterized costar modules and proved that the class of costar modules contains cotilting modules. In this paper similar results are obtained for Grothendieck categories. Let \(\mathbf F\colon\mathcal C\to\mathcal D\) and \(\mathbf G\colon\mathcal D\to\mathcal C\) be a pair of contravariant functors, additive and adjoint on the right, between Grothendieck categories \(\mathcal C\) and \(\mathcal D\) with \(U\) a generator of \(\mathcal C\) and \(V\) a reflexive generator of \(\mathcal D\). Then \(\mathbf F\) and \(\mathbf G\) define a duality between the subcategories \(\text{Copres}_{sf}(\mathbf G(V))\subseteq\mathcal C\) and \(V\)-\(fg\)-\(\text{Cogen}(\mathbf F(U))\subseteq\mathcal D\) which consist of objects of \(\mathcal C\) that are semifinitely copresented by \(\mathbf G(V)\) and the \(V\)-finitely generated objects of \(\mathcal D\) that are cogenerated by \(\mathbf F(U)\), respectively. – Some applications to graded rings are mentioned.

Reviewer: A. I. Kashu (Kishinev)

MSC:

16D90	Module categories in associative algebras
18E15	Grothendieck categories (MSC2010)
16E30	Homological functors on modules (Tor, Ext, etc.) in associative algebras
16W50	Graded rings and modules (associative rings and algebras)

Keywords:

dualities; Grothendieck categories; adjoint functors; graded rings; graded modules

Citations:

Zbl 0990.16009

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