Asymptotic stability of primes associated to homogeneous components of multigraded modules

Futoshi Hayasaka

doi:10.1016/j.jalgebra.2006.03.020

Asymptotic stability of primes associated to homogeneous components of multigraded modules

Futoshi Hayasaka

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

Let A ⊆ B be a homogeneous extension of Noetherian standard N^r-graded rings with A₀ = B₀ = R. Let M be a finitely generated N^r-graded B-module and N ⊆ M a finitely generated graded A-submodule of M. In this paper, we investigate the asymptotic behavior of the set of primes associated to the module M_n / N_n and prove that for all sufficiently large n ∈ N^r, the set Ass_R (M_n / N_n) is stable. We also give a certain inequality for the spread of standard multigraded rings, which is a natural generalization of Burch's inequality for the analytic spread of an ideal.

Original language	English
Pages (from-to)	535-543
Number of pages	9
Journal	Journal of Algebra
Volume	306
Issue number	2
DOIs	https://doi.org/10.1016/j.jalgebra.2006.03.020
Publication status	Published - Dec 15 2006
Externally published	Yes

Keywords

Associated prime
Grade
Multigraded
Spread

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jalgebra.2006.03.020

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abstract = "Let A ⊆ B be a homogeneous extension of Noetherian standard Nr-graded rings with A0 = B0 = R. Let M be a finitely generated Nr-graded B-module and N ⊆ M a finitely generated graded A-submodule of M. In this paper, we investigate the asymptotic behavior of the set of primes associated to the module Mn / Nn and prove that for all sufficiently large n ∈ Nr, the set AssR (Mn / Nn) is stable. We also give a certain inequality for the spread of standard multigraded rings, which is a natural generalization of Burch's inequality for the analytic spread of an ideal.",

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AB - Let A ⊆ B be a homogeneous extension of Noetherian standard Nr-graded rings with A0 = B0 = R. Let M be a finitely generated Nr-graded B-module and N ⊆ M a finitely generated graded A-submodule of M. In this paper, we investigate the asymptotic behavior of the set of primes associated to the module Mn / Nn and prove that for all sufficiently large n ∈ Nr, the set AssR (Mn / Nn) is stable. We also give a certain inequality for the spread of standard multigraded rings, which is a natural generalization of Burch's inequality for the analytic spread of an ideal.

KW - Associated prime

KW - Grade

KW - Multigraded

KW - Spread

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JO - Journal of Algebra

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ER -

Asymptotic stability of primes associated to homogeneous components of multigraded modules

Abstract

Keywords

ASJC Scopus subject areas

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