Torsion-free Modules Research Articles

Graded Bourbaki ideals of graded modules

In this paper we study graded Bourbaki ideals. It is a well-known fact that for torsionfree modules over Noetherian normal domains, Bourbaki sequences exist. We give criteria in terms of certain attached matrices for a homomorphism of modules to induce a Bourbaki sequence. Special attention is given to graded Bourbaki sequences. In the second part of the paper, we apply these results to the Koszul cycles of the residue class field and determine particular Bourbaki ideals explicitly. We also obtain in a special case the relationship between the structure of the Rees algebra of a Koszul cycle and the Rees algebra of its Bourbaki ideal.

Relative torsionfree modules with respect to a faithfully balanced bimodule

Let S and R be two associative rings, be a faithfully balanced (S, R)-bimodule. We introduce and investigate properties of ∞-torsionfree modules with respect to which are generalizations of n-torsionfree modules for any positive integer n and totally C-reflexive modules. And we give some characterizations of the class of the ∞- C-torsionfree modules over any ring R. Moreover, when C is a semidualizing (S, R)-bimodule, we study the relations between the classes of relative torsionfree modules and the torsionfree modules in terms of Foxby equivalence. Finally, as the ∞-C-torsionfree module which is in the left orthogonal class of C is exactly the C-Gorenstein projective module, we characterize the Auslander class with respect to a semidualizing module by virtue of the ∞-C-torsionfree modules over the commutative Cohen-Macaulay ring with a dualizing module, which is a module version of Theorem 4.6 in Holm, H., Jørgensen, P. (2006). [Semi-dualizing modules and related Gorenstein homologica dimensions. J.Pure Appl. Algebra 205(2):423–445].

On the vanishing of the theta invariant and a conjecture of Huneke and Wiegand

Huneke and Wiegand conjectured that, if $M$ is a finitely generated, non-free, torsion-free module with rank over a one-dimensional Cohen-Macaulay local ring $R$, then the tensor product of $M$ with its algebraic dual has torsion. This conjecture, if $R$ is Gorenstein, is a special case of a celebrated conjecture of Auslander and Reiten on the vanishing of self extensions that stems from the representation theory of finite-dimensional algebras. If $R$ is a one-dimensional Cohen-Macaulay ring such that $R=S/(f)$ for some local ring $(S, \mathfrak{n})$, and a non zero-divisor $f \in \mathfrak{n}^2$ on $S$, we make use of Hochster's theta invariant and prove that such $R$-modules $M$ which have finite projective dimension over $S$ satisfy the proposed torsion condition of the conjecture. Along the way we give several applications of our argument pertaining to torsion properties of tensor products of modules.

Arithmetic modules over generalized Dedekind domains

Let [Formula: see text] be a finitely generated torsion-free module over a generalized Dedekind domain [Formula: see text]. It is shown that if [Formula: see text] is a projective [Formula: see text]-module, then it is a generalized Dedekind module and [Formula: see text]-multiplication module. In case [Formula: see text] is Noetherian it is shown that [Formula: see text] is either a generalized Dedekind module or a Krull module. Furthermore, the polynomial module [Formula: see text] is a generalized Dedekind [Formula: see text]-module (a Krull [Formula: see text]-module) if [Formula: see text] is a generalized Dedekind module (a Krull module), respectively.

A Study of The Density Property in Module Theory

In this paper, there are two main objectives. The first objective is to study the relationship between the density property and some modules in detail, for instance; semisimple and divisible modules. The Addition complement has a good relationship with the density property of the modules as this importance is highlighted by any submodule N of M has an addition complement with Rad(M)=0. The second objective is to clarify the relationship between the density property and the essential submodules with some examples. As an example of this relationship, we studied the torsion-free module and its relationship with the essential submodules in module M.

Restricted Projective, Injective and Flat Complexes

We introduce and study the notions of restricted projective, injective and flat complexes of modules. It is shown in the paper that a complex C of modules is restricted projective if and only if for each $$m\in \mathbb {Z}$$ the module $$C^{m}$$ is restricted projective. A similar result for restricted injective complexes is also given. As an application, we characterize the complex of strongly torsion-free modules introduced by Xu.

A remark on prime ideals

If M is a torsion-free module over an integral domain, then we show that for each submodule N of M the envelope EM (N ) of N in M is an essential extension of N. In particular, if N is divisible then EM (N ) = N . The last condition says that N is a semiprime submodule of M if N is proper.Let M be a module over a ring R such that for any ideals a, b of R, (a ∩ b)M = aM ∩ bM . If N is an irreducible and weakly semiprime submodule of M , then we prove that (N :R M ) is a prime ideal of R. As a result, we obtain that if p is an irreducible ideal of a ring R such that a2 ⊆ p (a is an ideal of R) ⇒ a ⊆ p, then p is a prime ideal.

Finitely generated torsion-free modules over integrally closed domains

Let M be a finitely generated torsion-free module over a Dedekind domain R and be a polynomial module over where X is an indeterminate. It is shown that M and are both generalized Dedekind modules over R and respectively. These results are obtained using a relation between the v-submodules of M (v-submodules of ) and ideals of R (invertible ideals of ).Communicated by Toma Albu

Proper classes generated by τ- closed submodules

Abstract The main object of this paper is to study relative homological aspects as well as further properties of τ -closed submodules. A submodule N of a module M is said to be τ -closed (or τ -pure) provided that M/N is τ -torsion-free, where τ stands for an idempotent radical. Whereas the well-known proper class 𝒞losed (𝒫ure) of closed (pure) short exact sequences, the class τ −𝒞losed of τ -closed short exact sequences need not be a proper class. We describe the smallest proper class 〈τ − 𝒞losed〉 containing τ − 𝒞losed, through τ -closed submodules. We show that the smallest proper class 〈τ − 𝒞losed〉 is the proper classes projectively generated by the class of τ -torsion modules and coprojectively generated by the class of τ -torsion-free modules. Also, we consider the relations between the proper class 〈τ − 𝒞losed〉 and some of well-known proper classes, such as 𝒞losed, 𝒫ure.

Picard groups of some module categories

The groups of equivalence classes of auto-equivalences of the categories of torsion and torsion-free modules over an integral domain $R$ are determined. For the category of torsion-free modules, this group is isomorphic to the Picard group $\operatorname {Pic} R$ of $R$. We prove that for the category of torsion $R$-modules, this group is isomorphic to a group of certain divisible torsion groups called ``cyclones'' where addition is given in terms of the torsion product. In particular, this group is abelian and contains the direct product of the Picard group of the $R$-completion $\widetilde R$ of $R$ and of the group of ``clones'' of $Q/R$, where $Q$ denotes the quotient field of $R$. The structure of the latter group is described in terms of the inverse system of the unit groups $U(R/Rr)$ of the proper quotients $R/Rr$ ($r \in R$) of $R$.

Self-pure-generators over Dedekind domains

We prove that all pure submodules of a finite rank torsion-free module A over a Dedekind domain are A-generated (i.e. A is a self-pure-generator) if and only if A has a rank 1 direct summand B such that type(B) is the inner type of A. This result is applied to describe the direct products of torsion-free groups of finite rank which are self-pure-generators.

On the Ideal Case of a Conjecture of Huneke and Wiegand

AbstractA conjecture of Huneke and Wiegand claims that, over one-dimensional commutative Noetherian local domains, the tensor product of a finitely generated, non-free, torsion-free module with its algebraic dual always has torsion. Building on a beautiful result of Corso, Huneke, Katz and Vasconcelos, we prove that the conjecture is affirmative for a large class of ideals over arbitrary one-dimensional local domains. Furthermore, we study a higher-dimensional analogue of the conjecture for integrally closed ideals over Noetherian rings that are not necessarily local. We also consider a related question on the conjecture and give an affirmative answer for first syzygies of maximal Cohen–Macaulay modules.

Abelian varieties isogenous to a power of an elliptic curve over a Galois extension

Given an elliptic curve E/k and a Galois extension k ' /k, we construct an exact functor from torsion-free modules over the endomorphism ring EndE k ' with a semilinear Gal(k ' /k) action to abelian varieties over k that are k ' -isogenous to a power of E. As an application, we give a simple proof that every elliptic curve with complex multiplication geometrically is isogenous over the ground field to one with complex multiplication by a maximal order.

The symplectic structure on the moduli space of line bundles on a noncommutative Azumaya surface

In this note we prove that the moduli space of torsion-free modules of rank one over an Azumaya algebra on a K3 surface is an irreducible symplectic variety deformation equivalent to a Hilbert scheme of points on the K3 surface.

Auslander modules

In this paper, we introduce the notion of Auslander modules, inspired from Auslander's zero-divisor conjecture (theorem) and give some interesting results for these modules. We also investigate torsion-free modules.

On torsion free and cotorsion discrete modules

We prove that, if $\mathcal F $ is the class of torsion free discrete modules over a profinite group $G$, that is, the class of discrete $G$-modules which are torsion free as abelian groups, then $({\mathcal F},{\mathcal F}^\bot )$ is a complete cotorsion pair. Moreover, we find a structure theorem for torsion free and cotorsion discrete $G$-modules and for finitely generated cotorsion discrete $G$-modules.

The Algebra of Differential Operators for a Matrix Weight: An Ultraspherical Example

In this paper we study in detail algebraic properties of the algebra $\mathcal D(W)$ of differential operators associated to a matrix weight of Gegenbauer type. We prove that two second order operators generate the algebra, indeed $\mathcal D(W)$ is isomorphic to the free algebra generated by two elements subject to certain relations. Also, the center is isomorphic to the affine algebra of a singular rational curve. The algebra $\mathcal D(W)$ is a finitely-generated torsion-free module over its center, but it is not flat and therefore it is not projective. This is the second detailed study of an algebra $\mathcal D(W)$ and the first one coming from spherical functions and group representations. We prove that the algebras for different Gegenbauer weights and the algebras studied previously, related to Hermite weights, are isomorphic to each other. We give some general results that allow us to regard the algebra $\mathcal D(W)$ as the centralizer of its center in the Weyl algebra. We do believe that this should hold for any irreducible weight and the case considered in this paper represents a good step in this direction.

On $$\xi $$ ξ -torsion modules

In this paper we introduce the concept of $$\xi $$ -torsion module, $$\xi $$ -torsion-free module and $$\xi $$ -torsionable module. We investigate many properties of these modules. We characterize $$\xi $$ -torsion modules and $$\xi $$ -torsion-free modules using short exact sequences and module homomorphisms.

Picard groups and torsion-free cancellation for orders in $\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}$

We examine the question of direct-sum cancellation of finitely generated, torsion-free modules over ring orders $R$ contained in $\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}$. Using conditions involving the Picard group and the unit group of $R$, we give a nearly complete classification of those orders $R$ for which torsion-free cancellation holds. There is exactly one `exceptional' order to which our methods do not apply.

Torsion in tensor powers of modules

AbstractTensor products usually have nonzero torsion. This is a central theme of Auslander's 1961 paper; the theme continues in the work of Huneke and Wiegand in the 1990s. The main focus in this article is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modules N with the property that M ⊗R N has nonzero torsion unless M is very special. An important example of such a module N is the Frobenius power peR over a complete intersection domain R of characteristic p > 0.