The Gorenstein Homological Properties Of Category Algebras

We study the Gorensteinness of finite El category algebras,the tensor product of Gorenstein projective modules over finite EI category algebras,the maximal Cohen-Macaulay approximation of the trivial module over a finite El category algebra and the spectra of the singularity category of a Gorenstein category algebra.In Chapter 1,we briefly recall the historical orgins and current developments on category algebras,tensor triangulated categories,singularity categories and Gorenstein homological algebra.We introduce the main results.In Chapter 2,we introduce basic facts about category algebras,Gorenstein homo-logical algebra and upper triangular matrix rings.We explicitly characterize the projec-tive modules,the injective modules and the Gorenstein projective modules over upper triangular matrix rings.In Chapter 3,we recall basic properties of upper triangular matrix rings.We ob-serve that a finite El category algebra is isomorphic to a certain upper triangular matrix algebra.We introduce the notion of a projective El category.Then we prove that a fi-nite El category algebra is Gorenstein if and only if the given category is projective.We prove that a finite El category algebra is 1-Gorenstein if and only if the given category is a finite projective and free El category.In Chapter 4,we prove that Gorenstein projective modules over a finite El category algebra are closed under the tensor product if and only if its projective modules are closed under the tensor product;such a El category is called GPT-closed.We prove that if a finite projective El category is GPT-closed,then each morphism in the given category is a monomorphism.For a finite projective and free El category,we prove that it is GPT-closed if and only if each morphism is a monomorphism.In Chapter 5,we recall that the category of finitely generated modules over a finite El category algebra is equivalent to the category of covariant functors from the given finite El category to the category of finite dimensional vector spaces over the underlying field.For a finite free El category,we explicitly construct a functor E,and prove that the constructed functor E,viewed as a module over the category algebra,is a Gorenstein projective module if in addition the category is projective.We prove that the constructed functor E is a maximal Cohen-Macaulay approximation of the trivial module and the tensor identity of the stable category of Gorenstein projective modules over the category algebra.In Chapter 6,using Schur functors,we give a different proof of Fei Xu's result about the spectra of the bounded derived category of a category algebra via Verdier quotient functors.We describe the spectra of the singularity category of a Gorenstein category algebra.