Characterizations of injective cogenerators and Morita duality via equivalences and dualities (Q1901002)

Let FGP-\(R\) be the category of all finitely generated projective right \(R\)-modules and FCI-\(R\) the category of all finitely cogenerated injective right \(R\)-modules. The author proves the following main results: 1) The following statements are equivalent for a bimodule \(_AV_R\): i) \(V_R\) is a finitely cogenerated injective cogenerator and \(A\cong\text{End } V_R\) canonically; ii) \(-\oplus V :\text{FGP- }A\rightleftarrows\text{FCI-}R :\text{Hom}_R (_A V_R, -)\) defines an equivalence; iii) \(\text{Hom}_A(-,{_AV_R}) : A\text{- FGP}\rightleftarrows\text{FCI-}R :\text{Hom}_R (-,{_AV_R})\); 2) Let \(T :\text{FGP-}A\rightleftarrows\text{FCI-}P : H\) define an additive equivalence, and \(_A V_R = T(_A A_A)\) be the canonical bimodule. Then i) \(V_R\) is a finitely cogenerated injective cogenerator and \(A\cong\text{End} (V_R)\); ii) \(T\cong -\oplus_A V_R\) and \(H\cong\text{Hom}_R (_A V_R, -)\).