A note on the v-invariant

2023/12/28 by Aldo Conca, Conca, Aldo · 7 citations
Mathematics · #Commutative Algebra and Its Applications #Rings, Modules, and Algebras #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2401.00022

Abstract

Let R be a finitely generated \mathbb N-graded algebra domain over a Noetherian ring and let I be a homogeneous ideal of R. Given P∈ Ass(R/I) one defines the v-invariant v_P(I) of I at P as the least c∈ \mathbb N such that P=I:f for some f∈ R_c. A classical result of Brodmann asserts that Ass(R/Iⁿ) is constant for large n. So it makes sense to consider a prime ideal P∈ Ass(R/Iⁿ) for all the large n and investigate how v_P(Iⁿ) depends on n. We prove that v_P(Iⁿ) is eventually a linear function of n. When R is the polynomial ring over a field this statement has been proved independently also by Ficarra and Sgroi in a recent preprint.

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