Gorenstein Dimensions over Ring Homomorphisms

In this paper, we study injective dimension and Gorenstein injective dimension over local ring homomorphisms. Some well-known results are generalized. For example, the Bass formula for Gorenstein injective dimension of complexes is extended. As applications, some characterizations of Gorenstein rings are obtained.     

Gorenstein Dimension and AS-Gorenstein Algebras

The purpose of this paper is to connect the notion of Gorenstein dimension with AS-Gorenstein algebras. In particular, we show that a noetherian connected graded algebra having a balanced dualizing complex is AS-Gorenstein if the balanced dualizing complex has finite Gorenstein dimension. As a preparation, we generalize the Auslander–Bridger formula to the class of noncommutative noetherian connected graded algebras having balanced dualizing complexes.     

G-dimension over local homomorphisms with respect to a semi-dualizing complex

We study the G-dimension over local ring homomorphisms with respect to a semi-dualizing complex. Some results that track the behavior of Gorenstein properties over local ring homomorphisms under composition and decomposition are given. As an application, we characterize a dualizing complex for R in terms of the finiteness of the G-dimension over local ring homomorphisms with respect to a semi-dualizing complex.     

Cohen–Macaulay Isolated Singularities with a Dualizing Module

We generalize results of Foxby concerning a commutative Nötherian ring to a certain noncommutative Nötherian algebra Λ over a commutative Gorenstein complete local ring. We assume that Λ is a Cohen–Macaulay isolated singularity having a dualizing module. Then the same method as in the commutative cases works and we obtain a category equivalence between two subcategories of mod Λ, one of which includes all finitely generated modules of finite Gorenstein dimension. We give examples of such algebras which are not Gorenstien; orders related to almost Bass orders and some k-Gorenstein algebras for an integer k.Presented by I. Reiten ^★The author is supported by Grant-in-Aid for Scientific Researches B(1) No. 14340007 in Japan.     

Transfer of Gorenstein dimensions along ring homomorphisms

A central problem in the theory of Gorenstein dimensions over commutative noetherian rings is to find resolution-free characterizations of the modules for which these invariants are finite. Over local rings, this problem was recently solved for the Gorenstein flat and the Gorenstein projective dimensions; here we give a solution for the Gorenstein injective dimension. Moreover, we establish two formulas for the Gorenstein injective dimension of modules in terms of the depth invariant; they extend formulas for the injective dimension due to Bass and Chouinard.     

A Generalization of Auslander's Last Theorem

Before his death, Auslander announced that every finitely generated module over a local Gorenstein ring has a minimal Cohen–Macaulay approximation. Yoshimo extended Auslander's result to local Cohen–Macaulay rings admitting a dualizing module.Over a local Gorenstein ring the finitely generated maximal Cohen–Macaulay modules are the finitely generated Gorenstein projective modules so in fact Auslander's theorem says finitely generated modules over such rings have Gorenstein projective covers. We extend Auslander's theorem by proving that over a local Cohen–Macaulay ring admitting a dualizing module all finitely generated modules of finite G-dimension (in Auslander's sense) have a Gorenstein projective cover. Since all finitely generated modules over a Gorenstein ring have finite G-dimension, we recover Auslander's theorem when R is Gorenstein.     

Complete cohomology for complexes with finite Gorenstein AC-projective dimension

We study complete cohomology of complexes with finite Gorenstein AC-projective dimension. We show first that the class of complexes admitting a complete level resolution is exactly the class of complexes with finite Gorenstein AC-projective dimension. This lets us give some general techniques for computing complete cohomology of complexes with finite Gorenstein AC-projective dimension. As a consequence, the classical relative cohomology for modules of finite Gorenstein AC-projective dimension is extended. Finally, the relationships between projective dimension and Gorenstein AC-projective dimension for complexes are given.     

Complete intersection dimensions and Foxby classes

Let R be a local ring and M a finitely generated R-module. The complete intersection dimension of M-defined by Avramov, Gasharov and Peeva, and denoted -is a homological invariant whose finiteness implies that M is similar to a module over a complete intersection. It is related to the classical projective dimension and to Auslander and Bridger’s Gorenstein dimension by the inequalities .Using Blanco and Majadas’ version of complete intersection dimension for local ring homomorphisms, we prove the following generalization of a theorem of Avramov and Foxby: Given local ring homomorphisms φ:R→S and ψ:S→T such that φ has finite Gorenstein dimension, if ψ has finite complete intersection dimension, then the composition ψ°φ has finite Gorenstein dimension. This follows from our result stating that, if M has finite complete intersection dimension, then M is C-reflexive and is in the Auslander class A_C(R) for each semidualizing R-complex C.     

A Bass Formula for Gorenstein Injective Dimension

In this article a generalized version of the Bass formula is proved for finitely generated modules of finite Gorenstein injective dimension over a commutative Noetherian ring.     

Gorenstein projective and flat complexes over noetherian rings

We give sufficient conditions on a class of R‐modules $\mathcal {C}We give sufficient conditions on a class of R‐modules $\mathcal {C}$ in order for the class of complexes of $\mathcal {C}$‐modules, $dw \mathcal {C}$, to be covering in the category of complexes of R‐modules. More precisely, we prove that if $\mathcal {C}$ is precovering in R ? Mod and if $\mathcal {C}$ is closed under direct limits, direct products, and extensions, then the class $dw \mathcal {C}$ is covering in Ch(R). Our first application concerns the class of Gorenstein flat modules. We show that when the ring R is two sided noetherian, a complex C is Gorenstein flat if and only if each module C_n is Gorenstein flat. If moreover every direct product of Gorenstein flat modules is a Gorenstein flat module, then the class of Gorenstein flat complexes is covering. We consider Gorenstein projective complexes as well. We prove that if R is a commutative noetherian ring of finite Krull dimension, then the class of Gorenstein projective complexes coincides with that of complexes of Gorenstein projective modules. We also show that if R is commutative noetherian with a dualizing complex then every right bounded complex has a Gorenstein projective precover.     

On Gorenstein FP-Injective and Gorenstein Flat Complexes

In this article, the concept of Gorenstein FP-injective modules and some related known results are generalized to Gorenstein FP-injective complexes. Moreover, some new characterizations of Gorenstein flat complexes are given. It is also proved that every complex has a Gorenstein flat preenvelope over coherent rings with finite self-FP-injective dimension.     

Gorenstein projective dimension for complexes

We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology groups have a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension.

     

Gorenstein Homological Aspects of Monomorphism Categories via Morita Rings

In this paper we construct Gorenstein-projective modules over Morita rings with zero bimodule homomorphisms and we provide sufficient conditions for such rings to be Gorenstein Artin algebras. This is the first part of our work which is strongly connected with monomorphism categories. In the second part, we investigate monomorphisms where the domain has finite projective dimension. In particular, we show that the latter category is a Gorenstein subcategory of the monomorphism category over a Gorenstein algebra. Finally, we consider the category of coherent functors over the stable category of this Gorenstein subcategory and show that it carries a structure of a Gorenstein abelian category.     

Gorenstein Injective Dimension for Complexes and Iwanaga–Gorenstein Rings

The main purpose of this article is to present some applications of the notion of Gorenstein injective dimension of complexes over an associative ring. Among the applications, we give some new characterizations of Iwanaga–Gorenstein rings. In particular, we show that an associative ring R is Iwanaga–Gorenstein if and only if the class of complexes of Gorenstein injective dimension less than or equal to zero and the class of complexes of finite projective dimension are orthogonal complement of each other with respect to the ‘Ext’ functor.     

Lower complete intersection dimension over local homomorphisms

We introduce and study a theory of lower complete intersection dimension over local homomorphisms which encompasses the theory of lower complete intersection dimension for finite modules over local rings introduced by Gerko. In particular, we show that the lower complete intersection dimension over local homomorphisms reflects the complete intersection property of base rings as expected. As an application, we prove that the converse of a theorem of Sather-Wagstaff is also true.     

Rigid Dualizing Complexes Over Commutative Rings

In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. The method of rigidity was modified to work over general commutative base rings in our paper (Yekutieli and Zhang, Trans AMS 360:3211–3248, 2008). In the present paper we obtain many of the important local features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results apply to essentially finite type algebras over a regular noetherian finite dimensional base ring, and hence are suitable for arithmetic rings. In the sequel paper (Yekutieli, Rigid dualizing complexes on schemes, in preparation) these results will be used to construct and study rigid dualizing complexes on schemes. This research was supported by the US–Israel Binational Science Foundation. The second author was partially supported by the US National Science Foundation.     

Foxby duality and Gorenstein injective and projective modules

In 1966, Auslander introduced the notion of the -dimension of a finitely generated module over a Cohen-Macaulay noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander and Bridger (Mem. Amer. Math. Soc., vol. 94, 1969)). Enochs and Jenda attempted to dualize the notion of -dimensions. It seemed appropriate to call the modules with -dimension 0 Gorenstein projective, so the basic problem was to define Gorenstein injective modules. These were defined in Math. Z. 220 (1995), 611--633 and were shown to have properties predicted by Auslander's results. The way we define Gorenstein injective modules can be dualized, and so we can define Gorenstein projective modules (i.e. modules of -dimension 0) whether the modules are finitely generated or not. The investigation of these modules and also Gorenstein flat modules was continued by Enochs, Jenda, Xu and Torrecillas. However, to get good results it was necessary to take the base ring Gorenstein. H.-B. Foxby introduced a duality between two full subcategories in the category of modules over a local Cohen-Macaulay ring admitting a dualizing module. He proved that the finitely generated modules in one category are precisely those of finite -dimension. We extend this result to modules which are not necessarily finitely generated and also prove the dual result, i.e. we characterize the modules in the other class defined by Foxby. The basic result of this paper is that the two classes involved in Foxby's duality coincide with the classes of those modules having finite Gorenstein projective and those having finite Gorenstein injective dimensions. We note that this duality then allows us to extend many of our results to the original Auslander setting.

     

Cohomology theories for complexes

We introduce and study a complete cohomology theory for complexes, which provides an extended version of Tate–Vogel cohomology in the setting of (arbitrary) complexes over associative rings. Moreover, for complexes of finite Gorenstein projective dimension a notion of relative Ext is introduced. On the basis of these cohomology groups, some homological invariants of modules over commutative noetherian local rings, such as Martsinkovsky’s ξ-invariants and relative and Tate versions of Betti numbers, are extended to the framework of complexes with finite homology. The relation of these invariants with their prototypes is explored.     

On Gorenstein injective discrete modules over profinite groups

Gorenstein homological algebra was introduced in categories of modules. But it has proved to be a fruitful way to study various other categories such as categories of complexes and of sheaves. In this paper, the research of relative homological algebra in categories of discrete modules over profinite groups is initiated. This seems appropriate since (in some sense) the subject of Gorenstein homological algebra had its beginning with Tate homology and cohomology over finite groups. We prove that if the profinite group has virtually finite cohomological dimension then every discrete module has a Gorenstein injective envelope, a Gorenstein injective cover and we study various cohomological dimensions relative to Gorenstein injective discrete modules. We also study the connection between relative and Tate cohomology theories.     

群的Gorenstein同调维数

设$k$是一个弱维数有限的交换环, $G$是一个群. 本文讨论了群$G$具有有限的Gorenstein同调维数的标准.证明了群$G$的Gorenstein同调维数的有限性与群环$kG$的Gorenstein弱维数的有限性是一致的.进一步,我们给出了Serre定理的一个Gorenstein类比.推广了整环上$G$的Gorenstein同调维数的一些已知结果.