A note on the 𝑉-invariant

Abstract

Let R R be a finitely generated N \mathbb N -graded algebra domain over a Noetherian ring and let I I be a homogeneous ideal of R R . Given P ∈ A s s ( R / I ) P\in Ass(R/I) one defines the v v -invariant v P ( I ) v_P(I) of I I at P P as the least c ∈ N c\in \mathbb N such that P = I : f P=I:f for some f ∈ R c f\in R_c . A classical result of Brodmann [Proc. Amer. Math. Soc. 74 (1979), pp. 16–18] asserts that A s s ( R / I n ) Ass(R/I^n) is constant for large n n . So it makes sense to consider a prime ideal P ∈ A s s ( R / I n ) P\in Ass(R/I^n) for all the large n n and investigate how v P ( I n ) v_P(I^n) depends on n n . We prove that v P ( I n ) v_P(I^n) is eventually a linear function of n n . When R R is the polynomial ring over a field this statement has been proved independently also by Ficarra and Sgroi in their recent preprint [Asymptotic behaviour of the v \text {v} -number of homogeneous ideals, https://arxiv.org/abs/2306.14243, 2023].