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

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.

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)

Citations:

Zbl 0990.16009
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References:

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[2] DOI: 10.1016/S0007-4497(03)00043-5 · Zbl 1029.18003
[3] DOI: 10.1006/jabr.2001.8784 · Zbl 0990.16009
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