Gorenstein injective modules and Auslander categories

In this paper, we study Gorenstein injective modules over a local Noetherian ring R. For an R-module M, we show that M is Gorenstein injective if and only if Hom _R (Ȓ,M) belongs to Auslander category B(Ȓ), M is cotorsion and Ext ⁱ _R (E,M) = 0 for all injective R-modules E and all i > 0. Received: 24 August 2006 Revised: 30 October 2006     

Systèmes récursifs et algèbre de hadamard de suites récurrentes linéaires sur des anneaux commutatifs.

Abstract

Let 𝒢 be a simple finite dimensional Lie algebra over the complex numbers and let 𝒢¯ = 𝒢₁ ⊕…⊕ 𝒢 _k be a regular semisimple subalgebra of 𝒢 with each 𝒢 _i being a simple algebra of type A or C. It is shown that the lattice of submodules of a generalized Verma 𝒢-module constructed by parabolic induction starting from a simple torsion free 𝒢¯-module is almost always isomorphic to the lattice of submodules of an associated module formed as a quotient of a classical Verma module by a sum of Verma submodules. In particular, it is shown that the Mathieu admissible Verma modules involved have maximal submodules which are the sum of Verma modules.     

ON FLATNESS RELATIVE TO A TORSION THEORY

Abstract

For a torsion radical, σ, σ-flatness (as defined in [8]) is studied. Many of the properties of flat modules are shown to extend and σ-flat modules are characterized under certain restrictions on the associated filter, Lσ. We also define semi-σ-flat modules in a natural way and find conditions equivalent to every R-module being semi-σ-flat.     

On Modules Which are Isomorphic to Relatively Divisible or Pure Submodules of Each Other

Abstract

In this note, we provide a generalization of a well-known result of module theory which states that two injective modules are isomorphic when they are isomorphic to submodules of each other. More precisely, we show here that two RD-injective (respectively, pure-injective) modules over an integral domain are isomorphic if they are isomorphic to relatively divisible (respectively, pure) sub- modules of each other.     

A note on symbolic and ordinary powers of homogeneous ideals

In this note we are interested in the graded modulesM _k=I^(k)/I^k and , whereI is a saturated ideal in the homogeneous coordinate ringS=K[x₀,…,x_n] of ℙⁿ,I ^(k) is the symbolic power and is the saturation of the ordinary power. Very little is known about these modules, and we provide a bound on their diameters, we compute the Hilbert functions and we study some characteristic submodules in the special case ofn+1 general points in ℙⁿ.

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Sunto In questa nota siamo interessati ai moduli graduatiM _k=I(k)/I^k e , doveI è un ideale saturato nell'anello delle coordinate omogeneeS:=K[x₀,…,x_n] di ℙⁿ,I ^(k) è la potenza simbolica e è la saturazione della potenza ordinaria. Poco è noto su questi moduli e qui viene fornito un limite superiore ai loro diametri. Ne calcoliamo inoltre le funzioni di Hilbert e studiamo alcuni sottomoduli caratteristici nel caso speciale din+1 punti in posizione generale, in ℙⁿ.

     

Artinian level modules of embedding dimension two

We prove that a sequence of positive integers (h₀,h₁,…,h_c) is the Hilbert function of an artinian level module of embedding dimension two if and only if h_i−1−2h_i+h_i+1≤0 for all 0≤i≤c, where we assume that h₋₁=h_c+1=0. This generalizes a result already known for artinian level algebras. We provide two proofs, one using a deformation argument, the other a construction with monomial ideals. We also discuss liftings of artinian modules to modules of dimension one.     

Weak subintegral closure of ideals

We describe some basic facts about the weak subintegral closure of ideals in both the algebraic and complex-analytic settings. We focus on the analogy between results on the integral closure of ideals and modules and the weak subintegral closure of an ideal. We start by giving a new geometric interpretation of the Reid–Roberts–Singh criterion for when an element is weakly subintegral over a subring. We give new characterizations of the weak subintegral closure of an ideal. We associate with an ideal I of a ring A an ideal I_>, which consists of all elements of A such that v(a)>v(I), for all Rees valuations v of I. The ideal I_> plays an important role in conditions from stratification theory such as Whitney's condition A and Thom's condition Af and is contained in every reduction of I. We close with a valuative criterion for when an element is in the weak subintegral closure of an ideal. For this, we introduce a new closure operation for a pair of modules, which we call relative weak closure. We illustrate the usefulness of our valuative criterion.     

Vector fields and a family of linear type modules related to free divisors

This paper has three main goals. We start describing a method for computing the polynomial vector fields tangent to a given algebraic variety; this is of interest, for instance, in view of (effective) foliation theory. We then pass to furnishing a family of modules of linear type (that is, the Rees algebra equals the symmetric algebra), formed with vector fields related to suitable pairs of algebraic varieties, one of them being a free divisor in the sense of K. Saito. Finally, we derive freeness criteria for modules retaining a certain tangency feature, so that, in particular, well-known criteria for free divisors are recovered.     

Taylor complexes for Koszul boundaries

For all boundary modules of the Koszul complex of a monomial sequence we construct complexes, which we call Taylor complexes. For a monomial d-sequences these complexes provide free resolutions of the boundary modules. Let M be the ideal generated by a monomial d-sequence. We use the Taylor complexes to construct minimal free resolutions of the Rees algebra and the associated graded ring of M. Received: 13 November 1997 / Revised version: 6 March 1998     

A WEAKER FORM OF BAER'S SPLITTING PROBLEM OVER VALUATION DOMAINS

Abstract

Eklof-Fuchs [3] have shown that over an arbitrary valuation domain R, the modules B which satisfy Ext 1/R (B,T) = 0 for all torsion R-modules T are precisely the free R-modules. Here we modify the problem and describe all R-modules B for which Ext 1/R (B, T) vanishes for all bounded and for all divisible torsion R-modules T. It is well known that if R is a descrete rank one valuation domain then all torsion—free R-modules B have this property.     

Transversals of families in complete lattices,and torsion in product modules

Suppose L is a complete lattice containing no copy of the power-set 2 and no uncountable well-ordered chains. It is shown that for any family of nonempty subsets , one can choose elements p _i X _i so that _A p _i majorizes all elements of all but finitely many of the X _i . Ring-theoretic consequences are deduced: for instance, the direct product of a family of torsion modules over a commutative Noetherian integral domain R is torsion if and only if some element of R annihilates all but finitely many of the modules.     

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.     

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.     

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.     

Starke Kotorsionsmoduln

Suppose that $(R, m)$ is a noetherian local ring and that E is the injective hull of the residue class field $R/m$. Suppose that M is an R-module, $M^0 = {\mbox{\rm Hom}}_R (M, E)$ is the Matlis dual of M and ${\mbox{\rm Coass}(M)} = {\mbox{\rm Ass} (M^0)}$. M is called cotorsion if every prime ideal ${\frak p} \in {\mbox{\rm Coass}}(M)$ is regular; it is called strongly cotorsion if $\cap {\rm Coass}(M)$ is regular. In the first part, we completely describe the structure of the strongly cotorsion modules over R, use this to determine the coassociated prime ideals of the bidual $M^{00}$, and give in the second part criteria for a cotorsion module being strongly cotorsion. Received: 7 March 2002     

Bounds for indecomposable torsion-free modules

Let M be a finitely generated torsion-free module over a one-dimensional reduced Noetherian ring R with finitely generated normalization. The rank of M is the tuple of vector-space dimensions of M_P over each field R_P (R localized at P), where P ranges over the minimal prime ideals of R. We assume that there exists a bound N_R on the ranks of all indecomposable finitely generated torsion-free R-modules. For such rings, what bounds and ranks occur? Partial answers to this question have been given by a plethora of authors over the past forty years. In this article we provide a final answer by giving a concise list of the ranks of indecomposable modules for R a local ring with no condition on the characteristic. We conclude that if the rank of an indecomposable module M is (r,r,…,r), then r∈{1,2,3,4,6}, even when R is not local.     

Dominating sets of the comaximal and ideal-based zero-divisor graphs of commutative rings

Abstract

Let R be a commutative ring with nonzero identity, and let I be an ideal of R. The ideal-based zero-divisor graph of R, denoted by Γ_I (R), is the graph whose vertices are the set {x ∈ R \ I| xy ∈ I for some y∈ R \ I} and two distinct vertices x and y are adjacent if and only if xy ∈ I. Define the comaximal graph of R, denoted by CG(R), to be a graph whose vertices are the elements of R, where two distinct vertices a and b are adjacent if and only if Ra+Rb=R. A nonempty set S ? V of a graph G=(V, E) is a dominating set of G if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality among the dominating sets of G. The main object of this paper is to study the dominating sets and domination number of Γ_I (R) and the comaximal graph CG₂(R) \ J (R) (or CGJ (R) for short) where CG₂(R) is the subgraph of CG(R) induced on the nonunit elements of R and J (R) is the Jacobson radical of R.     

Self-dual modules over local rings of curve singularities

Let C be a reduced curve singularity. C is called of finite self-dual type if there exist only finitely many isomorphism classes of indecomposable, self-dual, torsion-free modules over the local ring of C. In this paper it is shown that the singularities of finite self-dual type are those which dominate a simple plane singularity.     

A note on some chain conditions

We answer a question raised by Othman Echi: Is an E ₁ (resp., a C ₁) ring an E (resp., a C) ring? We construct a C ₁ (thus E ₁) ring which is not an E ₂ (thus not a C ₂) ring. Received: 11 June 2007     

A new depth and the annihilation of local cohomology modules

Let J I be two proper ideals of a commutative Noetherian ring and M a finitely generated module. Strong relative depth is defined and characterized. It is proved that this depth is just the maximum integer n such that can be annihilated by some power of J for all i ≤ n. It turns out that the local-global principle for the annihilation of local cohomology modules can be formulated as a natural property of this new depth. Received: 28 July 2006