Eventually homological isomorphisms and Gorenstein projective modules

We prove that a certain eventually homological isomorphism between module categories induces triangle equivalences between their singularity categories,Gorenstein defect categories and stable categories of Gorenstein projective modules.Furthermore,we show that the Auslander-Reiten conjecture and the Gorenstein symmetry conjecture can be reduced by eventually homological isomorphisms.Applying these results to arrow removal and vertex removal,we describe the Gorenstein projective modules over some non-monomial algebras and verify the Auslander-Reiten conjecture for certain algebras.

Gorenstein projective modules and Frobenius extensions

We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective,then its underlying module over the base ring is Gorenstein projective; the converse holds if the frobenius extension is either left-Gorenstein or separable(e.g., the integral group ring extension ZZG).Moreover, for the Frobenius extension RA = R[x]/(x^2), we show that: a graded A-module is Gorenstein projective in GrMod(A), if and only if its ungraded A-module is Gorenstein projective, if and only if its underlying R-module is Gorenstein projective. It immediately follows that an R-complex is Gorenstein projective if and only if all its items are Gorenstein projective R-modules.

Derived representation type and Gorenstein projective modules of an algebra under crossed product

Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is an improvement of the conclusion about representation type of an algebra in Li and Zhang [Sci China Ser A, 2006, 50: 1-13]. Secondly, we give the relationship between Gorenstein projective modules over A and that over A#σH. Then, using this result, it is proven that A is a finite dimensional CM-finite Gorenstein algebra if and only if so is A#σH.

n-strongly Gorenstein Projective and Injective and Flat Modules

In this paper, we study some properties of n-strongly Gorenstein projective,injective and flat modules, and discuss some connections between n-strongly Gorenstein injective, projective and flat modules. Some applications are given.

Algebraic K-theory of Gorenstein projective modules

We introduce the Gorenstein algebraic K-theory space and the Gorenstein algebraic K-group of a ring, and show the relation with the classical algebraic K-theory space, and also show the ＇resolution theorem＇ in this context due to Quillen. We characterize the Gorenstein algebraic K-groups by two different Mgebraic K-groups and by the idempotent completeness of the Gorenstein singularity category of the ring. We compute the Gorenstein algebraic K-groups along a recollement of the bounded Gorenstein derived categories of CM-finite Gorenstein algebras.

Strongly Gorenstein Flat Modules and Dimensions

There is a variety of nice results about strongly Gorenstein flat modules over coherent rings. These results are done by Ding, Lie and Mao. The aim of this paper is to generalize some of these results, and to give homological descriptions of the strongly Gorenstein flat dimension (of modules and rings) over arbitrary associative rings.

Gorenstein Projective Dimensions of Complexes

We show that over a right coherent left perfect ring R, a complex C of left R-modules is Gorenstein projective if and only if C^m is Gorenstein projective in R-Mod for all m E Z. Basing on this we show that if R is a right coherent left perfect ring then Gpd（C） = sup{Gpd（C^m）｜m ∈ Z} where Gpd（-） denotes Gorenstein projective dimension.

Another Gorenstein Analogue of Projective Complexes

As we know,a complex P is projective if and only if P is exact with Z_n(P)projective in R-Mod for each n∈Z and any morphism f:P→C is null homotopic for any complex C.In this article,we study the notion of DG-Gorenstein projective complexes.We show that a complex G is DG-Gorenstein projective if and only if G is exact with Z_n(G)Gorenstein projective in R-Mod for each n∈Z and any morphism f:G→Q is null homotopic whenever Q is a DG-projective complex.

On Ding Projective Complexes

In the paper, Ding projective modules and Ding projective complexes are considered. In particular, it is proven that Ding projective complexes are precisely the complexes X for which each Xm is a Ding projective R-module for all m ∈ Z.

Acyclic Complexes and Gorenstein Rings

For a given class of modules A,let A be the class of exact complexes having all cycles in A,and dw(A)the class of complexes with all components in A.Denote by GL the class of Gorenstein injective modules.We prove that the following are equivalent over any ring R:every exact complex of injective modules is totally acyclic;every exact complex of Gorenstein injective modules is in every complex in dw(GL)is dg-Gorenstein injective.The analogous result for complexes of flat and Gorenstein flat modules also holds over arb计rary rings.If the ring is n-perfect for some integer n≥0,the three equivalent statements for flat and Gorenstein flat modules are equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules.We also prove the following characterization of Gorenstein rings.Let R be a commutative coherent ring;then the following are equivalent:(1)every exact complex of FP-injective modules has all its cycles Ding injective modules;(2)every exact complex of flat modules is F-totally acyclic,and every R-modulc M such that M^(+)is Gorenstein flat is Ding injective;(3)every exact complex of injectives has all its cycles Ding injective modules and every R-module M such that is Gorenstein flat is Ding injective.If R has finite Krull dimension,statements(1)-(3)are equivalent to(4)R is a Gorenstein ring(in the sense of Iwanaga).

Ding w-Flat Modules and Dimensions

The introduction of w-operation in the class of flat modules has been successful. Let R be a ring. An R-module M is called a w-fiat module if Tor1r（M, N） is GV-torsion for all R-modules N. In this paper, we introduce the w-operation in Gorenstein homological algebra. An R-module M is called Ding w-flat if there exists an exact sequence of projective R-modules ... → P1 → P0 → p0 → p1 → ... such that M Im（P0 → p0） and such that the functor HomR （-,F） leaves the sequence exact whenever F is w-flat. Several well- known classes of rings are characterized in terms of Ding w-flat modules. Some examples are given to show that Ding w-flat modules lie strictly between projective modules and Gorenstein projective modules. The Ding w-flat dimension （of modules and rings） and the existence of Ding w-flat precovers are also studied.

The stable Auslander-Reiten components of certain monomorphism categories

Let Λ be an Artin algebra and let Gprj-Λ denote the class of all the finitely generated Gorenstein projective Λ-modules. In this paper, we study the components of the stable Auslander-Reiten quiver of a certain subcategory of the monomorphism category S(Gprj-Λ) containing boundary vertices. We describe the shape of such components. It is shown that certain components are linked to the orbits of an auto-equivalence on the stable category Gprj. In particular, for the finite components, we show that under certain mild conditions,their cardinalities are divisible by 3. We see that this three-periodicity phenomenon reoccurs several times in the paper.

The Relative Transpose over Cohen-Macaulay Finite Artin Algebras

The relative transpose via Gorenstein projective modules is introduced,and some corresponding results on the Auslander-Reiten sequences and the Auslander-Reiten formula to this relative version are generalized.

Some Results on Noetherian Warfield Domains

Let R be a domain.In this paper,we show that if R is one dimensional,then R is a Noetherian Warfield domain if and only if every maximal ideal of R is 2-generated and for every maximal ideal M of R,M is divisorial in the ring(M:M).We also prove that a Noetherian domain R is a Noetherian Warfield domain if and only if for every maximal ideal M of R,M^(2) can be generated by two elements.Finally,we give a sufficient condition under which all ideals of R are strongly Gorenstein projective.