Multiplicities and Rees valuations

Katz, Daniel J.; Validashti, Javid

doi:10.1007/bf03191222

Cited by 24 publications

(15 citation statements)

References 16 publications

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“…The vanishing of the ε-multiplicity of an ideal is captured by the analytic spread of the ideal. Indeed, as in the case of j -multiplicity, the ε-multiplicity of I is not zero if and only if the analytic spread of I is maximal [23,41]. In particular, by Proposition 6.1 we obtain the following result.…”

Section: The ε -Multiplicity Of Edge Idealsmentioning

confidence: 72%

Generalized multiplicities of edge ideals

2017

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We explore connections between the generalized multiplicities of square-free monomial ideals and the combinatorial structure of the underlying hypergraphs using methods of commutative algebra and polyhedral geometry. For instance, we show the j -multiplicity is multiplicative over the connected components of a hypergraph, and we explicitly relate the j -multiplicity of the edge ideal of a properly connected uniform hypergraph to the Hilbert-Samuel multiplicity of its special fiber ring. In addition, we provide general bounds for the generalized multiplicities of the edge ideals and compute these invariants for classes of uniform hypergraphs.2010 Mathematics Subject Classification. 13H15, 13D40, 52B20.

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Section: The ε -Multiplicity Of Edge Idealsmentioning

confidence: 72%

Generalized multiplicities of edge ideals

2017

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“…Since the analytic spread of J is maximal we conclude from [20,Theorem 4.7] that the epsilon multiplicity ε R (J) = 0. This yields a contradiction.…”

Section: Multi-graded Hilbert Functionsmentioning

confidence: 83%

“…The following theorem may be compared with a nonvanishing result of Katz and Validashti [20,Theorem 4.7].…”

Section: Multi-graded Hilbert Functionsmentioning

confidence: 99%

Asymptotic growth of symbolic multi-Rees algebras

Das¹,

Roy²

2021

Preprint

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We study the question of finite generation of symbolic multi-Rees algebras and investigate the asymptotic behaviour of related length functions. In the setup of excellent local domains, we show that the symbolic multi-Rees algebra of a finite collection of ideals is finitely generated when the analytic spread is not maximal and the associated length function is eventually a polynomial. We further prove that the symbolic multi-Rees algebra of monomial ideals is finitely generated with no assumptions on the analytic spread. In this case, the corresponding length function is shown to exhibit piecewise quasi-polynomial behaviour.Proposition 4.1. [13, Lemma 3.4] There exists a vector h ∈ N r such that Ann R M n = Ann R M h for all n ≥ h. In particular, dim M n = dim M h for all n ≥ h.

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“…For related work in a more general setting the reader is referred to D. Katz and J. Validashti [8]. Proof.…”

Section: Degree Function Coefficients Of I (Not Necessarily Simple)mentioning

confidence: 99%

On the degree function coefficients of complete ideals in two-dimensional regular local rings

Debremaeker

2013

Journal of Pure and Applied Algebra

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a b s t r a c tLet I be a complete M-primary ideal in a two-dimensional regular local ring (R, M) with an algebraically closed residue field. Let v 1 , . . ., v n denote the Rees valuations of I. In the theory of degree functions Rees and Sharp have associated with I positive integers d(I, v 1 ), . . ., d(I, v n ) providing useful information about I. We have called these integers the degree function coefficients of I. Using Zariski's theory of complete ideals in a two-dimensional regular local ring and recent work of Heinzer and Kim on complete ideals, we obtain new interpretations of the degree function coefficients of I. Some applications are given.

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Multiplicities and Rees valuations

Cited by 24 publications

References 16 publications

Generalized multiplicities of edge ideals

Generalized multiplicities of edge ideals

Asymptotic growth of symbolic multi-Rees algebras

On the degree function coefficients of complete ideals in two-dimensional regular local rings

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