Homological Liftings In Comma Categories And Extension Dimensions

In this paper,A and B are abelian categories,Λ and Γ are artin algebras.Our main results are as follows.In Chapter 2,let F:A→B be an additive and right exact functor which is perfect,and let(F,B)be the left comma category.We give an equivalent characterization of Gorenstein projective objects in(F,B)in terms of Gorenstein projective objects in B and A.We prove that there exists a left recollement of the stable category of the subcategory of(F,B)consisting of Gorenstein projective objects modulo projectives relative to the same kind of stable categories in B and A.Moreover,this left recollement can be filled into a recollement when B is Gorenstein and F preserves projectives.In Chapter 3,let(?)be the triangular matrix algebra with M a finitely generated(Λ,Γ)-bimodule.We construct support τ-tilting modules and(τ-)tilting modules in mod T from that in mod Λ and mod Γ,and give the converse constructions under some condition.In Chapter 4,we prove that the radical layer length of Λ is an upper bound for the radical layer length of mod Λ.We give an upper bound for the extension dimension of mod Λ in terms of the injective dimension of a certain class of simple left Λ-modules and the radical layer length of DΛ.