Relative Homological Dimensions And Recollements

In this paper,∧’,∧ and ∧" are artin algebras.We mainly study the following four aspects:In Chapter 2,we introduce and study(pre)resolving subcategories of a triangulated category and the homological dimensions relative to these subcategories.Then we apply the obtained results to relative Gorenstein categories.In Chapter 3,we prove that if there is a recollement of the bounded Gorenstein derived category Dgb(P(Mod ∧))(Mod ∧)relative to the bounded Gorenstein derived cat-egories Dgb(P(Mod ∧’)(Mod ∧’)and Dgb(P(Mod ∧"))(Mod ∧"),then A is Gorenstein if and only if so are ∧’and ∧".In addition,we prove that a virtually Gorenstein algebra A is Gorenstein if and only if the bounded homotopy category of(finitely generat-ed)projective left A-modules and that of(finitely generated)injective left A-modules coincide.Let(A,B,C)be a recollement of abelian categories.In Chapter 4,we show that torsion pairs in A and C can induce torsion pairs in B;and the converse holds true under certain conditions.Let(mod ∧’,mod ∧,mod ∧")be a recollement of categories of finitely generated left modules.In Chapter 5,we give a construction of gluing of tilting modules in mod ∧ with respect to tilting modules in mod ∧’ and mod ∧" as well as the converse construction.