Purity And Recollements In Gorenstein Homological Theory

Homological methods, in particular, relative homological algebra and derived cat-egories, are important tools in the study of Gorenstein algebras and related modules. In this thesis, we investigate the pure exact and recollement structures on Gorenstein projective and injective modules.This thesis is divided into three chapters.In Chapter 1, we state main results and list the symbols and notions used through-out the thesis.In Chapter 2, we introduce and study (weak) pure-injective Gorenstein projective modules. Let R be an artin algebra. We prove that the category of weak pure-injective Gorenstein projective left R-modules coincides with the intersection of the category of pure-injective left R-modules and that of Gorenstein projective left R-modules. Then, we get an equivalent characterization of virtually Gorenstein algebras (being CM-finite). Furthermore, we prove that the category of weak pure-injective Gorenstein projective left.R-modules is enveloping in the category of left R-modules; and if R is virtually Gorenstein, then it is precovering in the category of pure-injective left.R-modules.In Chapter 3, we first give an equivalence between the derived category of a locally finitely presented category and the derived category of contravariant functors from its finitely presented subcategory to the category of abelian groups, in the spirit of H. Krause's work. Then we provide a criterion for the existence of recollement of derived categories of functor categories, which shows that the recollement structure may be induced by a proper morphism defined in finitely presented subcategories. This criterion is then used to construct a recollement of derived categories of Gorenstein injective modules over CM-finite 2-Gorenstein artin algebras.