Contravariant functors on the category of finitely presented modules. (Q1001414)

By \(R\)-mod is denoted the category of finitely presented left \(R\)-modules. In this article a theory is developed for the category \(((R\text{-mod})^{\text{op}},\text{Ab})\) of contravariant functors, where \(R\) is an arbitrary ring, that is analogous to the theory of Auslander and Reiten (1973) for the category \(\text{fp}((\Lambda\text{-mod})^{\text{op}},\text{Ab})\) of finitely presented contravariant functors when the ring is an Artin algebra \(\Lambda\). An important problem in this domain is the existence of minimal flat resolutions in the category \(((R\text{-mod})^{\text{op}},\text{Ab})\). It is proved that such resolutions exist and they are characterized by pure-injective left \(R\)-modules and pure morphisms. The functor \(F\colon(R\text{-mod})^{\text{op}}\to\text{Ab}\) is cotorsion if \(\text{Ext}((-,N),F)=0\) for every flat object \((-,N)\) of \(((R\text{-mod})^{\text{op}},\text{Ab})\). For a left \(R\)-module \(M\) it is shown that the flat contravariant functor \((-,M)\) is cotorsion if and only if \(M\) is pure-injective. This fact is used to characterize when a flat resolution of an object \(F\in((R\text{-mod})^{\text{op}},\text{Ab})\) is minimal and to construct a minimal flat resolution of \(F\), given a projective presentation. Another important result is characterization of the situation when the category \(((R\text{-mod})^{\text{op}},\text{Ab})\) is locally Noetherian (if and only if for every module \(M\) the object \(\text{Ext}^1(-,M)\) is injective in \(((R\text{-mod})^{\text{op}},\text{Ab}))\). A description of the contravariant Gabriel spectrum of \(R\) is obtained (i.e., the set of indecomposable injective objects of the functor category \(((R\text{-mod})^{\text{op}},\text{Ab}))\). The points are in bijective correspondence with the set of pure-injective indecomposable left \(R\)-modules, which correspond to the points of the covariant Gabriel spectrum of \(R\). This paper contains also many other results and a series of interesting examples.