One-dimensional Local Rings Research Articles

Some remarks on two-periodic modules over local rings

In this paper, some properties of finitely generated two-periodic modules over commutative Noetherian local rings have been studied. We show that under certain assumptions on a pair of modules [Formula: see text] with [Formula: see text] being two-periodic, the natural map [Formula: see text] is an isomorphism. As a consequence, we prove that Auslander’s depth formula holds for such a Tor-independent pair. Tor-independence plays a crucial role for the depth formula to hold. Under certain assumptions on the modules, we show that a pair of modules, over a one-dimensional local ring, is Tor-independent if and only if their tensor product is torsion-free. Celikbas et al. recently showed the Huneke–Wiegand conjecture holds for two-periodic modules over one-dimensional domains. We generalize their result to the case of two-periodic modules with rank over one-dimensional local rings.

Generalized Loewy length of Cohen-Macaulay local and graded rings

We generalize a theorem of Ding relating the generalized Loewy length gℓℓ(R) and index of a one-dimensional Cohen-Macaulay local ring (R,m,k). Ding proved that if R is Gorenstein, the associated graded ring is Cohen-Macaulay, and k is infinite, then the generalized Loewy length and index of R are equal. However, if k is finite, equality may not hold. We prove that if the index of a one-dimensional Cohen-Macaulay local ring is finite and the associated graded ring has a homogeneous nonzerodivisor of degree t, then gℓℓ(R)≤index(R)+t−1. Next we prove that if R is a one-dimensional hypersurface ring with a witness to the generalized Loewy length that induces a regular initial form on the associated graded ring, then the generalized Loewy length achieves this upper bound. We then compute the generalized Loewy lengths of several families of examples of one-dimensional hypersurface rings over finite fields. Finally, we study a graded version of the generalized Loewy length and determine its value for numerical semigroup rings.

Reflexive modules over the endomorphism algebras of reflexive trace ideals

In the present paper we investigate reflexive modules over the endomorphism algebras of reflexive trace ideals in a one-dimensional Cohen-Macaulay local ring. The main theorem generalizes both of the results of S. Goto, N. Matsuoka, and T. T. Phuong ([20, Theorem 5.1]) and T. Kobayashi ([30, Theorem 1.3]) concerning the endomorphism algebra of its maximal ideal. We also explore the question of when the category of reflexive modules is of finite type, i.e., the base ring has only finitely many isomorphism classes of indecomposable reflexive modules. We show that, if the category is of finite type, the ring is analytically unramified and has only finitely many Ulrich ideals. As a consequence, Arf local rings contain only finitely many Ulrich ideals once the normalization is a local ring.

On one-dimensional local rings and Berger’s conjecture

AbstractLet k be a field of characteristic zero and let $\Omega_{A/k}$ be the universally finite differential module of a k-algebra A, which is the local ring of a closed point of an algebraic or algebroid curve over k. A notorious open problem, known as Berger’s Conjecture, predicts that A must be regular if $\Omega_{A/k}$ is torsion-free. In this paper, assuming the hypotheses of the conjecture and observing that the module ${\rm Hom}_A(\Omega_{A/k}, \Omega_{A/k})$ is then isomorphic to an ideal of A, say $\mathfrak{h}$, we show that A is regular whenever the ring $A/a\mathfrak{h}$ is Gorenstein for some parameter a (and conversely). In addition, we provide various characterizations for the regularity of A in the context of the conjecture.

A Hilbert–Kunz function with a periodic term that has a given period

A result of Monsky states that the Hilbert–Kunz function of a one-dimensional local ring of prime characteristic has a term ϕ \phi that is eventually periodic. For example, in the case of a power series ring in one variable over a prime-characteristic field, ϕ \phi is the zero function and is therefore immediately periodic with period 1. In additional examples produced by Kunz [Amer. J. Math. 91 (1969), pp. 772–784] and Monsky [Math. Ann. 263 (1983), pp. 43–49], ϕ \phi is immediately periodic with period 2. We show that, for every positive integer π \pi , there exists a ring for which ϕ \phi is immediately periodic with period π \pi .

When is $${\mathfrak {m}}:{\mathfrak {m}}$$ an almost Gorenstein ring?

Given a one-dimensional Cohen-Macaulay local ring $$(R,{\mathfrak {m}},k)$$ , we prove that it is almost Gorenstein if and only if $${\mathfrak {m}}$$ is a canonical module of the ring $${\mathfrak {m}}:{\mathfrak {m}}$$ . Then, we generalize this result by introducing the notions of almost canonical ideal and gAGL ring and by proving that R is gAGL if and only if $${\mathfrak {m}}$$ is an almost canonical ideal of $${\mathfrak {m}}:{\mathfrak {m}}$$ . We use this fact to characterize when the ring $${\mathfrak {m}}:{\mathfrak {m}}$$ is almost Gorenstein, provided that R has minimal multiplicity. This is a generalization of a result proved by Chau, Goto, Kumashiro, and Matsuoka in the case in which $${\mathfrak {m}}:{\mathfrak {m}}$$ is local and its residue field is isomorphic to k.

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.

The first Brauer–Thrall conjecture over one-dimensional local rings

Let [Formula: see text] be a one-dimensional commutative Noetherian complete local ring. Assume that the cohomology annihilator [Formula: see text] is [Formula: see text]-primary. We use the notion of Gabriel–Roiter (co)measure in the category of maximal Cohen–Macaulay [Formula: see text]-modules and prove that, if there is an infinite set [Formula: see text] of indecomposable maximal Cohen–Macaulay [Formula: see text]-modules of bounded multiplicity, then there are indecomposable maximal Cohen–Macaulay modules of arbitrary large multiplicity which are cogenerated by modules in [Formula: see text]. This, in particular, guarantees the validity of the first Brauer–Thrall-type theorem for the category of maximal Cohen–Macaulay [Formula: see text]-modules.

Trace ideals, normalization chains, and endomorphism rings

In this paper we consider reduced (non-normal) commutative noetherian rings R.With the help of conductor ideals and trace ideals of certain R-modules we deduce a criterion for a reflexive R-module to be closed under multiplication with scalars in an integral extension of R. Using results of Greuel and Knorrer this yields a characterization of plane curves of finite Cohen–Macaulay type in terms of trace ideals. Further, we study one-dimensional local rings (R,m) such that that their normalization is isomorphic to the endomorphism ring EndR(m): we give a criterion for this property in terms of the conductor ideal, and show that these rings are nearly Gorenstein. Moreover, using Grauert–Remmert normalization chains, we show the existence of noncommutative resolutions of singularities of low global dimensions for curve singularities.

Rigid Ideals in Gorenstein Rings of Dimension One

Motivated by a conjecture of Huneke and Wiegand concerning torsion in tensor products of modules over local rings, we investigate the existence of ideals I in a one-dimensional Gorenstein local ring R satisfying $\text {Ext}^{1}_{R}(I,I)= 0$ .

Generators of reductions of ideals in a local Noetherian ring with finite residue field

Let ( R , m ) (R,\mathfrak {m}) be a local Noetherian ring with residue field k k . While much is known about the generating sets of reductions of ideals of R R if k k is infinite, the case in which k k is finite is less well understood. We investigate the existence (or lack thereof) of proper reductions of an ideal of R R and the number of generators needed for a reduction in the case k k is a finite field. When R R is one-dimensional, we give a formula for the smallest integer n n for which every ideal has an n n -generated reduction. It follows that in a one-dimensional local Noetherian ring every ideal has a principal reduction if and only if the number of maximal ideals in the normalization of the reduced quotient of R R is at most | k | |k| . In higher dimensions, we show that for any positive integer, there exists an ideal of R R that does not have an n n -generated reduction and that if n ≥ dim ⁡ R n \geq \dim R this ideal can be chosen to be m \mathfrak {m} -primary. In the case where R R is a two-dimensional regular local ring, we construct an example of an integrally closed m \mathfrak {m} -primary ideal that does not have a 2 2 -generated reduction and thus answer in the negative a question raised by Heinzer and Shannon.

The Structure of Chains of Ulrich Ideals in Cohen-Macaulay Local Rings of Dimension One

This paper studies Ulrich ideals in one-dimensional Cohen-Macaulay local rings. A correspondence between Ulrich ideals and overrings is given. Using the correspondence, chains of Ulrich ideals are closely explored. The specific cases where the rings are of minimal multiplicity and \operatorname{GGL} rings are analyzed.

A uniform general Neron desingularization in dimension one

We give a uniform General Neron Desingularization (GND) for one-dimensional local rings with respect to morphisms which coincide modulo a high power of the maximal ideal. The result has interesting applications in the case of Cohen–Macaulay rings.

One-dimensional Gorenstein local rings with decreasing Hilbert function

In this paper we solve a problem posed by M.E. Rossi: Is the Hilbert function of a Gorenstein local ring of dimension one not decreasing? More precisely, for any integer h>1, h∉{14+22k,35+46k|k∈N}, we construct infinitely many one-dimensional Gorenstein local rings, included integral domains, reduced and non-reduced rings, whose Hilbert function decreases at level h; moreover, we prove that there are no bounds to the decrease of the Hilbert function. The key tools are numerical semigroup theory, especially some necessary conditions to obtain decreasing Hilbert functions found by the first and the third author, and a construction developed by V. Barucci, M. D'Anna and the second author, that gives a family of quotients of the Rees algebra. Many examples are included.

One-dimensional stable rings

A commutative ring R is stable provided every ideal of R containing a nonzerodivisor is projective as a module over its ring of endomorphisms. The class of stable rings includes the one-dimensional local Cohen–Macaulay rings of multiplicity at most 2, as well as certain rings of higher multiplicity, necessarily analytically ramified. The former are important in the study of modules over Gorenstein rings, while the latter arise in a natural way from generic formal fibers and derivations.We characterize one-dimensional stable local rings in several ways. The characterizations involve the integral closure R‾ of R and the completion of R in a relevant ideal-adic topology. For example, we show: If R is a reduced stable ring, then there are exactly two possibilities for R: (1) R is a Bass ring, that is, R is a reduced Noetherian local ring such that R‾ is finitely generated over R and every ideal of R is generated by two elements; or (2) R is a bad stable domain, that is, R is a one-dimensional stable local domain such that R‾ is not a finitely generated R-module.

On a conjecture by Wilf about the Frobenius number

Given coprime positive integers $a_1 < ...< a_d$, the Frobenius number $F$ is the largest integer which is not representable as a non-negative integer combination of the $a_i$. Let $g$ denote the number of all non-representable positive integers: Wilf conjectured that $d \geq \frac{F+1}{F+1-g}$. We prove that for every fixed value of $\lceil \frac{a_1}{d} \rceil$ the conjecture holds for all values of $a_1$ which are sufficiently large and are not divisible by a finite set of primes. We also propose a generalization in the context of one-dimensional local rings and a question on the equality $d = \frac{F+1}{F+1-g}$.

Almost Gorenstein rings – towards a theory of higher dimension

The notion of almost Gorenstein local ring introduced by V. Barucci and R. Fröberg for one-dimensional Noetherian local rings which are analytically unramified has been generalized by S. Goto, N. Matsuoka and T.T. Phuong to one-dimensional Cohen–Macaulay local rings, possessing canonical ideals. The present purpose is to propose a higher-dimensional notion and develop the basic theory. The graded version is also posed and explored.

Wang's theorem for one-dimensional local rings

Wang's theorem for one-dimensional local rings

The Frobenius functor and injective modules

We investigate commutative Noetherian rings of prime characteristic such that the Frobenius functor applied to any injective module is again injective. We characterize the class of one-dimensional local rings with this property and show that it includes all one-dimensional F F -pure rings. We also give a characterization of Gorenstein local rings in terms of T o r i R ( R f , E ) \mathrm {Tor}_i^R(R^{f},E) , where E E is the injective hull of the residue field and R f R^{f} is the ring R R whose right R R -module action is given by the Frobenius map.

On the Buchsbaumness of the Associated Graded Ring of a One-Dimensional Local Ring

Let (R, m) be a Noetherian, one-dimensional, local ring, with |R/m|=∞. We study when its associated graded ring G(m) is Buchsbaum; in particular, we give a theoretical characterization for G(m) to be Buchsbaum not Cohen–Macaulay. Finally, we consider the particular case of R being the semigroup ring associated to a numerical semigroup S: we introduce some invariants of S, and we use them in order to give a necessary and a sufficient condition for G(m) to be Buchsbaum.