On F-almost split sequences

Let . be an Artinian algebra and F an additive subbifunctor of Ext 1.(.,.) having enough projectives and injectives. We prove that the dualizing subvarieties of mod . closed under F-extensions have F-almost split sequences. Let T be an F-cotilting module in mod . and S a cotilting module over . = End(T). Then Hom(., T) induces a duality between F-almost split sequences in .F T and almost split sequences in . S, where add. S = Hom.( P (F), T). Let . be an F-Gorenstein algebra, T a strong F-cotilting module and 0 . A . B . C . 0 an F-almost split sequence in . F T. If the injective dimension of S as a gT-module is equal to d, then C . (. .dCM.d DTrA*)*, where (-)* = Hom(-, T). In addition, if the F-injective dimension of A is equal to d A...dCMFD..dFopTRC...dCMF.dFDTrC.