Reflexive modules over Arf local rings

Isobe, Ryotaro; Kumashiro, Shinya

doi:10.48550/arxiv.2105.07184

Cited by 2 publications

(4 citation statements)

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“…Thereafter, Lipman [29] generalized them by extracting the essence of the rings. The reader can consult [21,29] for several basic results on Arf rings. The examples arising from numerical semigroup rings are also found by [30,Corollary 3.19].…”

Section: Observation Let R = K[[h]mentioning

confidence: 99%

“…Then, Lindo proved that the endomorphism algebra T (M) = Hom R (tr R (M), tr R (M)) is the center of Hom R (M, M) ( [26,Introduction]). It is also known that M can be regarded as a T (M)-module ( [3, (7.2) Proposition], [21,Proposition 2.4]). If tr T (M ) M = T (M) and T (M) is a local ring, then T (M) is a direct summand of M. Recently, Isobe and Kumashiro, and independently Dao, provided a certain direct-sum decomposition for reflexive modules over (one-dimensional) Arf local rings by using these results ([6, Theorem A] and [21,Theorem 1.1]).…”

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: Observation Let R = K[[h]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

When are trace ideals finite?

Kumashiro¹

2021

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“…Since (M ∨ ) * is a reflexive module, we can regard (M ∨ ) * as an R-module (see [15,Proposition 2.4] or [1, (7.2) Proposition]). Since R is a discrete valuation ring, it follows that (M ∨ ) * ∼ = (R) r .…”

Section: Corollary 27 Suppose That R Satisfies the Conditions (A)-(d)...mentioning

confidence: 99%

“…3 and e(R 1 ) = 5. (ii) Let R 2 = K[|t 9 , t 10 , t 11 , t 12 , t 15 |]. Then R 2 is a far-flung Gorenstein ring with r(R 2 ) = 4 and e(R 1 ) = 9.…”

Section: We Obtain the Assertion (Ii)mentioning

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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Reflexive modules over Arf local rings

Cited by 2 publications

References 14 publications

When are trace ideals finite?

When are trace ideals finite?

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

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