On a Finitistic Cotilting-type Duality

Let R and S be arbitrary associative rings. Given a bimodule RWS , we denote by ∆? and Γ? the functors Hom?(−, W ) and Ext?(−, W ), where ? = R or S. We say that RWS is a finitistic weakly cotilting bimodule (briefly FWC) if for each module M cogenerated by W , finitely generated or homomorphic image of a finite direct sum of copies of W , ΓM = 0 = Ext(M, W ). We are able to describe, on a large class of finitely generated modules, the cotilting-type duality induced by a FWC-bimodule. Introduction Let R and S be arbitrary associative rings with unity and RWS be a R-S-bimodule. We denote by ∆R and ∆S the contravariant functors HomR(−,W ) : R-Mod→ Mod-S and HomS(−,W ) : Mod-S → R-Mod and by ΓR and ΓS the contravariant functors ExtR(−,W ) : R-Mod→ Mod-S and ExtS(−,W ) : Mod-S → R-Mod. In this paper we are interested to describe a cotilting-type duality on the categories of finitely generated modules. The notion of cotilting module appears, in the context of finitely generated modules over artin algebras, as dual of the concept of tilting module, introduced in the early eighties by Brenner and Butler [2] and Happel and Ringel [9]. In 1989 Colby in [3] extended the notion of cotilting to finitely generated modules over noetherian rings which are noetherian over their endomorphism ring. Next, the notion has been once more extended and studied for arbitrary modules over arbitrary rings (see [4, 6, 8, 5, 7]). In this paper we reconsider the original spirit of the theory concentrating on objects with finiteness conditions, but over arbitrary associative rings. The idea was inspired by reading the paper [1] of Angeleri Hügel. She studied dualities induced by a balanced Colby bimodule RWS (see [1, section 2]), i.e. a bimodule satisfying the following conditions: 1. Ext?(X,W ) = 0 for all finitely presented modules X, 2. Ext?(W,W ) = 0, 3. if Hom?(X,W ) = 0 = Ext?(X,W ) and X is finitely presented, then M = 0, 4. RW and WS are finitely generated and the functors ∆? carry finitely generated modules to finitely generated modules, for ? = R and S. For such a bimodule, the classes Y? of finitely generated modules cogenerated by W coincide with the classes of finitely presented modules belonging to Ker Γ? (see [1, Proposition 2.6, 4.3]); the functors ∆? induce a duality between the classes Y? and the functors Γ? induce a duality between the classes X ′ ? of finitely presented modules in Ker∆? (see [1, Theorem 4.4]). A bimodule RWS is called weakly cotilting (see [10]) if the left Rand right Smodules cogenerated byW are contained in Ker ΓR and Ker ΓS , respectively, and the injective dimension of RW and WS is less than or equal to 1. We say that RWS is a finitistic weakly cotilting bimodule (briefly FWC-bimodule) if for each module M cogenerated by W , finitely generated or homomorphic image of a finite direct sum of copies of W , we have ΓM = 0 = Ext(M,W ). For what we have observed before about 1