Gorenstein Projective Module Categories And Monomorphism Categories

Since1960's, Gorenstein homological algebra is a very successful theory of relativehomological algebra. It has wide applications in representation theory, Tate cohomol-ogy, singularity theory, etc. The basic idea of Gorenstein homological algebra is thatwe get Gorenstein projective modules, Gorenstein injective modules and Gorensteinfat modules instead of projective modules, injective modules and fat modules, re-spectively. For Artin algebras, the fundamental of Gorenstein homological algebrais Gorenstein projective modules, since we can get Gorenstein injective modules byapplying Nakayama functor on Gorenstein projective modules. In order to better un-derstand and develop Gorenstein homological algebra, we must construct nontrivialGorenstein projective modules and study the category of Gorenstein projective mod-ules more specifcally. In the topic of constructing nontrivial Gorenstein projectivemodules, there have been some breakthrough results given by Li and Zhang, it stillneed further development. On the other hand, the monomorphism category, as a gen-eralization of submodule categories, is closely related with the categories of Gorensteinprojective modules and singularity categories. It is important for studying the cate-gory of Gorenstein projective modules, because it gives a nice study framework for thecategories of Gorenstein projective modules over many algebras.Our main results are the following.In Chapter3, we study the construction of Gorenstein projective modules overtriangular matrix Artin algebras. We get the following:(1) For a very big class of triangular matrix algebras, we give the specifc construc-tion of the modules in the left Ext-orthogonal category of regular module.(2) For a great class of triangular matrix Gorenstein algebras, we give the specifcconstruction of Gorenstein projective Λ-modules.(3) We inductively give a class of Artin algebras of fnite Cohen-Macaulay typeby using triangular matrix extensions.In Chapter4, we generalize the Auslander-Reiten theory of submodule categoriesto monomorphism categories, and apply it to the Gorenstein projective module cate-gories of Tn-extensions of selfnjective algebras. We get the following:(1) We give a computable formula of the Auslander-Reiten translation of monomor-phism category from the Auslander-Reiten translation of the given algebra.(2) For the monomorphism category of a selfnjective algebra, namely, the cate- gory of Gorenstein projective modules over the Tn-extension of the selfnjectivealgebra, we give an expression of the power of the Auslander-Reiten transla-tion on objects. In particular, we give a specifc formula of the periodicity forNakayama selfnjective algebras.(3) For the stable monomorphism category of a selfnjective algebra, we give anexpression of the power of Serre functor on objects. For Nakayama selfnjectivealgebras, we give a specifc formula of the periodicity.(4) Finally, we give Auslander-Reiten quivers of some monomorphism categoriesof fnite representation type.In Chapter5, we introduce and study Gorenstein stable equivalence. We get thefollowing:(1) We show that the Gorenstein stable module category is the quotient categoryof the stable module category relative to the stable category of Gorensteinprojective modules.(2) For fnite dimensional algebras over a feld, we show that both the stableequivalence of Morita type and the almost ν-stable derived equivalence induceGorenstein stable equivalence.(3) We also give some examples to show the diferences among stable equiva-lence, Gorenstein stable equivalence, the equivalence of the stable category ofGorenstein projective modules and other equivalences.In Chapter6, for connected left artinian ring with radical square zero, we explic-itly describe the left Ext-orthogonal category of the left regular module, and give anefective bound for the generalized Nakayama conjecture.