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.     

An extension of a theorem of Frobenius and Stickelberger to modules of projective dimension one over a factorial domain

Let R be a Cohen–Macaulay ring. A quasi-Gorenstein R-module is an R-module such that the grade of the module and the projective dimension of the module are equal and the canonical module of the module is isomorphic to the module itself. After discussing properties of finitely generated quasi-Gorenstein modules, it is shown that this definition allows for a characterization of diagonal matrices of maximal rank over a Cohen–Macaulay factorial domain R extending a theorem of Frobenius and Stickelberger to modules of projective dimension 1 over a commutative factorial Cohen–Macaulay domain.     

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.     

Classifying thick subcategories of the stable category of Cohen-Macaulay modules

Various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher-dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite p-group due to Benson, Carlson and Rickard, we consider classifying thick subcategories of the stable category of Cohen-Macaulay modules over a Gorenstein local ring. The main result of this paper yields a complete classification of the thick subcategories of the stable category of Cohen-Macaulay modules over a local hypersurface in terms of specialization-closed subsets of the prime ideal spectrum of the ring which are contained in its singular locus.     

Flat modules over valuation rings

Let R be a valuation ring and let Q be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if Q is maximal (respectively artinian). It is shown that each singly projective module is a content module if and only if any non-unit of R is a zero-divisor and that each singly projective module is locally projective if and only if R is self-injective. Moreover, R is maximal if and only if each singly projective module is separable, if and only if any flat content module is locally projective. Necessary and sufficient conditions are given for a valuation ring with non-zero zero-divisors to be strongly coherent or π-coherent.A complete characterization of semihereditary commutative rings which are π-coherent is given. When R is a commutative ring with a self-FP-injective quotient ring Q, it is proved that each flat R-module is finitely projective if and only if Q is perfect.     

The homotopy category of complexes of projective modules

The homotopy category of complexes of projective left-modules over any reasonably nice ring is proved to be a compactly generated triangulated category, and a duality is given between its subcategory of compact objects and the finite derived category of right-modules.     

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.     

Two theorems about maximal Cohen–Macaulay modules

This paper contains two theorems concerning the theory of maximal Cohen–Macaulay modules. The first theorem proves that certain Ext groups between maximal Cohen–Macaulay modules M and N must have finite length, provided only finitely many isomorphism classes of maximal Cohen–Macaulay modules exist having ranks up to the sum of the ranks of M and N. This has several corollaries. In particular it proves that a Cohen–Macaulay lo cal ring of finite Cohen–Macaulay type has an isolated singularity. A well-known theorem of Auslander gives the same conclusion but requires that the ring be Henselian. Other corollaries of our result include statements concerning when a ring is Gorenstein or a complete intersection on the punctured spectrum, and the recent theorem of Leuschke and Wiegand that the completion of an excellent Cohen–Macaulay local ring of finite Cohen–Macaulay type is again of finite Cohen–Macaulay type . The second theorem proves that a complete local Gorenstein domain of positive characteristic p and dimension d is F-rational if and only if the number of copies of R splitting out of divided by has a positive limit. This result relates to work of Smith and Van den Bergh. We call this limit the F-signature of the ring and give some of its properties. Received: 6 May 2001 / Published online: 6 August 2002 Both authors were partially supported by the National Science Foundation. The second author was also partially supported by the Clay Mathematics Institute.     

Quasi-injective dimension

Following our previous work about quasi-projective dimension [11], in this paper, we introduce quasi-injective dimension as a generalization of injective dimension. We recover several well-known results about injective and Gorenstein-injective dimensions in the context of quasi-injective dimension such as the following. (a) If the quasi-injective dimension of a finitely generated module M over a local ring R is finite, then it is equal to the depth of R. (b) If there exists a finitely generated module of finite quasi-injective dimension and maximal Krull dimension, then R is Cohen-Macaulay. (c) If there exists a nonzero finitely generated module with finite projective dimension and finite quasi-injective dimension, then R is Gorenstein. (d) Over a Gorenstein local ring, the quasi-injective dimension of a finitely generated module is finite if and only if its quasi-projective dimension is finite.     

On the vanishing of (co)homology over local rings

Considering modules of finite complete intersection dimension over commutative Noetherian local rings, we prove (co)homology vanishing results in which we assume the vanishing of nonconsecutive (co)homology groups. In fact, the (co)homology groups assumed to vanish may be arbitrarily far apart from each other.     

IG-projective modules

This paper concerns finitely generated modules over Artin algebras. We introduce the notion of an IG-projective module and use this to prove that if, over such an algebra R$R$, each simple module is strongly Gorenstein projective, then any indecomposable R$R$-module is either projective or simple. We also prove that if R$R$ is local and the simple module is IG-projective, then 1-self-orthogonal modules are projective.     

Two criteria to check whether ideals are direct sums of cyclically presented modules

A famous theorem of commutative algebra due to I. M. Isaacs states that “if every prime ideal of R is principal, then every ideal of R is principal”. Therefore, a natural question of this sort is “whether the same is true if one weakens this condition and studies rings in which ideals are direct sums of cyclically presented modules?” The goal of this paper is to answer this question in the case R is a commutative local ring. We obtain an analogue of Isaacs's theorem. In fact, we give two criteria to check whether every ideal of a commutative local ring R is a direct sum of cyclically presented modules, it suffices to test only the prime ideals or structure of the maximal ideal of R. As a consequence, we obtain: if R is a commutative local ring such that every prime ideal of R is a direct sum of cyclically presented R-modules, then R is a Noetherian ring. Finally, we describe the ideal structure of commutative local rings in which every ideal of R is a direct sum of cyclically presented R-modules.     

Virtually Gorenstein rings and relative homology of complexes

We extend the notion of virtually Gorenstein rings to the setting of arbitrary rings, and prove that all rings R of finite Gorenstein weak global dimension are virtually Gorenstein such that all Gorenstein projective R-modules are Gorenstein flat. For such a ring R, we introduce the notion of relative homology functors of complexes with respect to Gorenstein projective (resp., flat) modules, and establish a balanced and a vanishing result for the homology functor.     

On the vanishing and the finiteness of supports of generalized local cohomology modules

Let be a Noetherian local ring, I an ideal of R and M, N two finitely generated R-modules. The first result of this paper is to prove a vanishing theorem for generalized local cohomology modules which says that for all j > dim(R), provided M is of finite projective dimension. Next, we study and give characterizations for the least and the last integer r such that Supp is infinite. This work is supported in part by the National Basis Research Programme in Natural Science of Vietnam.     

Gorenstein dimension of modules over homomorphisms

Given a homomorphism of commutative noetherian rings R→S and an S-module N, it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals , where E is the injective hull of the residue field of R. This result is analogous to a theorem of André on flat dimension.     

Local cohomology modules with respect to an ideal containing the irrelevant ideal

Let R=?_n≥0R_n be a homogeneous Noetherian ring, let M be a finitely generated graded R-module and let R₊=?_n>0R_n. Let b?b₀+R₊, where b₀ is an ideal of R₀. In this paper, we first study the finiteness and vanishing of the n-th graded component of the i-th local cohomology module of M with respect to b. Then, among other things, we show that the set becomes ultimately constant, as n→−∞, in the following cases:

(i)

and (R₀,m₀) is a local ring;

(ii)

dim(R₀)≤1 and R₀ is either a finite integral extension of a domain or essentially of finite type over a field;

(iii)

i≤g_b(M), where g_b(M) denotes the cohomological finite length dimension of M with respect to b.

Also, we establish some results about the Artinian property of certain submodules and quotient modules of .     

Rank-sets of indecomposable modules over one-dimensional rings

Let R be a one-dimensional, reduced Noetherian ring with finite normalization, and suppose there exists a positive integer N_R such that, for every indecomposable finitely generated torsion-free R-module M and every minimal prime ideal P of R, the dimension of M_P, as a vector space over the localization R_P (a field), is less than or equal to N_R. For a finitely generated torsion-free R-module M, we call the set of all such vector-space dimensions the rank-set of M. What subsets of the integers arise as rank-sets of indecomposable finitely generated torsion-free R-modules? In this article, we give more information on rank-sets of indecomposable modules, to supplement previous work concerning this question. In particular we provide examples having as rank-sets those intervals of consecutive integers that are not ruled out by an earlier article of Arnavut, Luckas and Wiegand. We also show that certain non-consecutive rank-sets never arise.     

The countable Telescope Conjecture for module categories

By the Telescope Conjecture for Module Categories, we mean the following claim: “Let R be any ring and (A,B) be a hereditary cotorsion pair in Mod-R with A and B closed under direct limits. Then (A,B) is of finite type.”We prove a modification of this conjecture with the word ‘finite’ replaced by ‘countable.’ We show that a hereditary cotorsion pair (A,B) of modules over an arbitrary ring R is generated by a set of strongly countably presented modules provided that B is closed under unions of well-ordered chains. We also characterize the modules in B and the countably presented modules in A in terms of morphisms between finitely presented modules, and show that (A,B) is cogenerated by a single pure-injective module provided that A is closed under direct limits. Then we move our attention to strong analogies between cotorsion pairs in module categories and localizing pairs in compactly generated triangulated categories.     

Mixed multiplicities and the minimal number of generator of modules

Let (R,m) be a d-dimensional Noetherian local ring. In this work we prove that the mixed Buchsbaum-Rim multiplicity for a finite family of R-submodules of R^p of finite colength coincides with the Buchsbaum-Rim multiplicity of the module generated by a suitable superficial sequence, that is, we generalize for modules the well-known Risler-Teissier theorem. As a consequence, we give a new proof of a generalization for modules of the fundamental Rees’ mixed multiplicity theorem, which was first proved by Kirby and Rees in (1994, [8]). We use the above result to give an upper bound for the minimal number of generators of a finite colength R-submodule of R^p in terms of mixed multiplicities for modules, which generalize a similar bound obtained by Cruz and Verma in (2000, [5]) for m-primary ideals.