(Strong) Gorenstein-projective Modules Over Morita Rings

Gorenstein algebra and Gorenstein-projective modules are important research topics in relative homological algebra,representation theory of algebras,triangulated categories,algebraic geometry(especially in singularity theory).For a given algebra A,how to judge whether it is a Gorenstein algebra,and how to construct all the Gorenstein-projective A-modules,are two fundamental problems in Gorenstein homological algebra.For algebras A and B,bimodules BMA and ANB,and a B-B-bimodule map φ:M(?)A N→ B,and an A-A-bimodule map ψ:N(?)B M→A satisfying some special conditions,Hyman Bass has introduced in his famous Mimeographed note Morita algebra Δ(φ,ψ)=Δ(φ,ψ)(A,B,M,N)=(BMAA ABNB),where the special conditions for φ andψ are to guarantee that the multiplication ofΔ(φ,ψ)(A,B,M.N)has the associativity.Morita algebras Δ(φ,ψ)(A,B,M,N)give a very large class of algebras,and many important algebras can be realized as Morita algebras.For examples,the 2 × 2 matrix algebra M2(A)=(AAAA)over A,the algebra Δ(0,0)(A)=(AAAA),the upper triangular matrix algebra(A0 ANBB),the algebras defined by finite quivers and relations.This thesis for Ph.D.is to study the Gorensteinness,Gorenstein-projective modules,strongly complete projective resolutions,and strong Gorenstein-projective modules,of Morita ringsΔ(φ,ψ)(A,B,M,N).The main contents are as follows.The first chapter is an introduction.We survey the relevant backgrounds and progress in the field we are working on,and we present the main results in this thesis.In the second chapter,we fix some notations,recall some necessary basic notion and results on Morita rings,Gorenstein-projective modules,strongly complete projective resolutions,strong Gorenstein-projective modules,and Gorenstein algebras,which will be used in this thesis.Chapter 3 is devoted to the study of the Gorensteinness of Morita rings.We prove the following two results:IfΔ(φ,ψ)(A,B,M,N)is a self-injective algebra,MA and AN(respectively,NB and BM)are projective modules,then A(respectively,B)is a self-injective algebra.IfΔ(φ,ψ)(A,B,M,N)is a Gorenstein algebra,MA and AN(respectively,NB and BM)are of finite projective dimension,then A(respectively,B)is a Gorenstein algebra.As a consequence,we get that Δ(φ,χ)(A)=(AAAA)is a self-injective algebra if and only if A is a self-injective algebra.In Chapter 4,we investigate relations between Gorenstein-projective modules over a Morita ring Δ(φ,ψ)(A,B,M,N)=(BMAA ANBB),and Gorenstein-projective modules over A and B.We give sufficient conditions such that the functor TA:A-mod→Δ(φ,ψ)-mod(respectively,TB:B-mod→ Δ(φ,ψ)-mod)preserves Gorenstein-projective modules.Given a Gorenstein-projective module(X,Y,f,g)over a Morita ring Δ(φ,ψ)(A,B,M,N),we also give a sufficient condition such that AX is a Gorenstein-projective A-module,and BY is a Gorenstein-projective B-module.Also,we get sufficient conditions such that the CM-freeness and CM-finiteness of Morita ring Δ(φ,ψ)(A,B,M,N)can be inherited by A and B.In the final chapter of this thesis,we obtain a necessary and sufficient condition for all the strongly complete projective resolutions over Morita ring Δ(0,0)(A,B,M,N)=(BMAA ANBB).As special cases,we get a class of strongly complete projective resolutions over Δ(0,0)(A,B,M,N),from the ones over A and B.We obtain a necessary and sufficient condition for all the strong Gorenstein-projective modules over Morita ring Δ(0.0)(A,B,M,N).As special cases,we get a class of strong Gorenstein-projective modules over Δ(0.0)(A,B,M,N),from the ones over A and B.This in particular applies to the algebra Δ(0,0)(A)=(AAAA).We also determine all the strongly complete projective resolutions and all the strong Gorenstein-projective modules,over the 2 × 2 matrix algebra M2(A)over A.