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Natural dualities. (English) Zbl 1069.16011

Summary: Let \(S\) be an arbitrary associative ring and \(_SW\) be a left \(S\)-module. Denote by \(R\) the ring \(\text{End}_SW\) and by \(\Delta\) both the contravariant functors \(\operatorname{Hom}_S(-,W)\) and \(\operatorname{Hom}_R(-,W)\). A module \(M\) is reflexive if the evaluation map \(\delta_M\colon M\to\Delta^2M\) is an isomorphism. Any direct summand of finite direct sums of copies of \(_SW\) and \(R_R\) is reflexive. Increasing in a minimal way the classes of reflexive modules, a “cotilting condition” on finitely generated \(R\)-modules naturally arises.

MSC:

16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16D90 Module categories in associative algebras
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