On reflexive and $I$-Ulrich modules over curve singularities

Dao, Hailong; Maitra, Sarasij; Sridhar, Prashanth

doi:10.48550/arxiv.2101.02641

Cited by 3 publications

(7 citation statements)

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“…By Fact 2.1(b), (d) and as mentioned in the introduction, trace ideals have deep connections with direct-sum decompositions of (reflexive) modules. With this observation, we can expect that the finiteness of trace ideals is related to the finiteness of indecomposable reflexive modules (e.g., see [9,Proposition 7.9]).…”

Section: When Are Ideals Trace Ideals?mentioning

confidence: 99%

“…On the other hand, Herzog and Rahimbeigi proved that for one-dimensional analytically irreducible Gorenstein local K-algebras, where K is an infinite field, the finiteness of trace ideals is equivalent to the finiteness of indecomposable maximal Cohen-Macaulay modules ( [19,Corollary 2.16]). Other progresses on trace ideals are seen in, for examples, [7,8,9,10,11,15,17,18,20,22,23,27]. Among them, in this paper, we study the following finiteness problem on trace ideals, which is also posed by several papers [9,Question 7.16(1)], [10,Question 3.7], and [19].…”

Section: Introductionmentioning

confidence: 99%

“…Other progresses on trace ideals are seen in, for examples, [7,8,9,10,11,15,17,18,20,22,23,27]. Among them, in this paper, we study the following finiteness problem on trace ideals, which is also posed by several papers [9,Question 7.16(1)], [10,Question 3.7], and [19].…”

Section: Introductionmentioning

confidence: 99%

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When are trace ideals finite?

Kumashiro¹

2021

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In this paper, we study Noetherian local rings having finite trace ideals. We proved that such rings are of dimension at most one. For the onedimensional case, we can reduce the situation that rings are Cohen-Macaulay. Then, we give a necessary condition to have finite trace ideals in terms of the value set obtained by the canonical module. We also gave the correspondence between trace ideals of R and that of the endomorphism algebra of the maximal ideal of R when R has minimal multiplicity.

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Section: When Are Ideals Trace Ideals?mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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When are trace ideals finite?

Kumashiro¹

2021

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“…Let R be a local Cohen-Macaulay ring and let Ref R be the category of finitely generated reflexive modules over R. This paper is motivated by the recent work on Ref R in [9]. In particular, we aim to answer the following question raised there: Question 1.1.…”

Section: Introductionmentioning

confidence: 99%

Reflexive modules, self-dual modules and Arf rings

Dao

2021

Preprint

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We prove a tight connection between reflexive modules over a one-dimensional ring R and its birational extensions that are self-dual as R-modules. Consequently, we show that a complete local reduced Arf ring has finitely many indecomposable reflexive modules up to isomorphism, which can be represented precisely by the local rings infinitely near it. We also characterize Arf rings by the property that any reflexive module is self-dual. We give applications on dimension of subcategories and existence of endomorphism rings of small global dimension (non-commutative desingularizations). Our results indicate striking similarities between Arf rings in dimension one and rational singularities in dimension two from representation-theoretic and categorical perspectives.

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“…Let R = K[|t4 , t 5 , t6 |] be a numerical semigroup ring, where K is a field. Let m be the maximal ideal of R. Then R is not far-flung Gorenstein by Example 2.9.…”

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confidence: 99%

The tiny trace ideals of the canonical modules in Cohen-Macaulay rings of dimension one

Herzog¹,

Kumashiro²,

Stamate³

2021

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We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We investigate far-flung Gorenstein rings in relation with the endomorphism algebras of the maximal ideals and numerical semigroup rings. We show that the solution of the Rohrbach problem in additive number theory provides an upper bound for the multiplicity of far-flung Gorenstein numerical semigroup rings. Reflexive modules over far-flung Gorenstein rings are also studied. 2020 Mathematics Subject Classification. 13H10. Key words and phrases. trace ideal, canonical module, Cohen-Macaulay ring, Gorenstein ring. The second author was supported by JSPS KAKENHI Grant Number 21K13766. Far-flung Gorenstein rings and bounds of the multiplicityThroughout this paper, unless otherwise noted, let (R, m) be a Cohen-Macaulay local ring of dimension one, possessing the canonical module ω R . We set the conditions on R as follows. Conditions. (a)There exists an R-submodule C of Q(R) such that R ⊆ C ⊆ R and C ∼ = ω R . (b) R is finitely generated as an R-module. (c) R is a local ring with the maximal ideal n. (d) There exists a non-zerodivisor a ∈ m such that (a) is a reduction of m.

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On reflexive and $I$-Ulrich modules over curve singularities

Abstract: We study reflexive modules over one dimensional Cohen-Macaulay rings.Our key technique exploits the concept of I-Ulrich modules.

Cited by 3 publications

References 15 publications

When are trace ideals finite?

When are trace ideals finite?

Reflexive modules, self-dual modules and Arf rings

The tiny trace ideals of the canonical modules in Cohen-Macaulay rings of dimension one

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