Variation of Hilbert coefficients

Ghezzi, Laura; Gotō, Shirō; Hong, Jooyoun; Vasconcelos, Wolmer V.

doi:10.1090/s0002-9939-2013-11774-0

Cited by 4 publications

(5 citation statements)

References 19 publications

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“…Then I ∼ = K R and mI = mt 2a −a−1 , where m denotes the maximal ideal of R. Hence R is an almost Gorenstein local ring (see[11, Example 2.13] for details). 2…”

mentioning

confidence: 99%

Almost Gorenstein rings – towards a theory of higher dimension

Gotō

Takahashi

Taniguchi

2015

Journal of Pure and Applied Algebra

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109

153

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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.

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“…Then I ∼ = K R and mI = mt 2a −a−1 , where m denotes the maximal ideal of R. Hence R is an almost Gorenstein local ring (see[11, Example 2.13] for details). 2…”

mentioning

confidence: 99%

Almost Gorenstein rings – towards a theory of higher dimension

Gotō

Takahashi

Taniguchi

2015

Journal of Pure and Applied Algebra

Self Cite

109

153

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“…where e 0 (I) is the multiplicity of I and o(I) its order ideal, that is the smallest positive integer n such that I ⊂ m n . A non-Cohen-Macaulay version of the similar character is given in [8,Theorem 3.3] with e 0 (I) replaced by λ(R/Q) for any reduction Q of I.…”

Section: Introductionmentioning

confidence: 99%

“…If is Cohen–Macaulay, we have from [29, Theorem 2.45] where is the multiplicity of and is the order of ; that is, the largest positive integer such that . A non-Cohen–Macaulay version of similar character is given in [8, Theorem 3.3], with replaced by for any reduction of . Since , by Serre’s theorem [2, Theorem 4.7.10] and [9, Theorem 7.2] respectively, the reduction number can be bounded in terms of alone.…”

Section: Introductionmentioning

confidence: 99%

Sally Modules and Reduction Numbers Of ideals

Ghezzi¹,

Gotō²,

Hong³

et al. 2016

Nagoya Math. J.

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Abstract. We study the relationship between the reduction number of a primary ideal of a local ring relative to one of its minimal reductions and the multiplicity of the corresponding Sally module. This paper is focused on three goals: (i) To develop a change of rings technique for the Sally module of an ideal to allow extension of results from Cohen-Macaulay rings to more general rings. (ii) To use the fiber of the Sally modules of almost complete intersection ideals to connect its structure to the CohenMacaulayness of the special fiber ring. (iii) To extend some of the results of (i) to two-dimensional Buchsbaum rings. Along the way we provide an explicit realization of the S 2 -fication of arbitrary Buchsbaum rings.

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“…As will be seen, some relationships involve the Hilbert coefficients f 0 (I) and f 1 (I) of the special fiber. We will follow te discussion of [54].…”

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confidence: 98%

“…Let (R, m) be a Cohen-Macaulay local ring of dimension d and infinite residue field. For an m-primary ideal I,red(I) ≤ d • λ(R/J) o(I) − 2d + 1,where J is a minimal reduction of I and o(I) is the m-adic order of I.To establish such a result for arbitrary Noetherian rings ([54]), we proceed differently. The version of the following lemma for Cohen-Macaulay rings can be found in[164, Chapter 3, Theorem 1.1].…”

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confidence: 99%

Complexity Degrees of Algebraic Structures

Vasconcelos

2014

Preprint

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We aim at studying collections C(R) of algebraic structures defined over a commutative ring R and investigating the complexity of significant constructions carried out on these objects. Noteworthy are the category of finitely generated modules (over local rings, or graded, particularly vector bundles), finitely generated algebras or objects suitable of more fundamental decompositions and construction of various closure operations-such as the determination of global sections and the (theoretical or machine) computation of integral closures of algebras. Typically, the operations express smoothing processes that enhance the structure of the algebras, enable them to support new constructions (including analytic ones), divisors acquire a group structure, and the cohomology tends to slim down. The assignment of measures of size, via a multiplicity theory, to the algebras and to the construction itself is a novel aspect to the subject. One of its specific goals is to develop a comprehensive theory of normalization in commutative algebra and a broad set of multiplicity functions to be used as complexity benchmarks.

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Variation of Hilbert coefficients

Cited by 4 publications

References 19 publications

Almost Gorenstein rings – towards a theory of higher dimension

Almost Gorenstein rings – towards a theory of higher dimension

Sally Modules and Reduction Numbers Of ideals

Complexity Degrees of Algebraic Structures

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