On F-almost split sequences

Abstract

Let Λ be an Artinian algebra and F an additive subbifunctor of Ext ¹_Λ (−,−) having enough projectives and injectives. We prove that the dualizing subvarieties of mod Λ closed under F-extensions have F-almost split sequences. Let T be an F-cotilting module in mod Λ and S a cotilting module over Γ = End(T). Then Hom(−, T) induces a duality between F-almost split sequences in ^⊥F T and almost split sequences in ^⊥ S, where add_Γ S = Hom_Λ(

(F), T). Let Λ 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 ≅ (Ω ^−d _CM Ω^d DTrA*)*, where (-)* = Hom(-, T). In addition, if the F-injective dimension of A is equal to d $A \cong \Omega _{CM_F }^{ - d} D\Omega _{F^{op} }^{ - d} TRC \cong \Omega _{CM_F }^{ - d} \Omega _F^d DTrC$.