A note on the $v$-invariant
Aldo Conca
TL;DR
The paper studies the asymptotic behavior of the $v$-invariant $v_P(I^n)$ for primes $P$ in the set of asymptotic associated primes ${\mathcal A}(I)$ of a homogeneous ideal $I$ in a finitely generated $\mathbb{N}$-graded domain $R$. It defines $v_P(I)$ via a minimal degree condition and employs a Rees algebra framework, constructing a finitely generated bigraded module $H$ to connect colon-ideals with graded components. The main result is that $v_P(I^n)$ becomes linear in $n$ for large $n$, with leading coefficient drawn from the degrees $d_1,\dots,d_c$ of the generators of $I$, extending known results in the polynomial ring case. The approach clarifies the relationship between asymptotic prime behavior and the graded structure of $I$, offering a concrete method to compute these invariants and highlighting the role of Rees-algebra techniques in asymptotic invariants.
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 $P\in Ass(R/I)$ one defines the $v$-invariant $v_P(I)$ of $I$ at $P$ as the least $c\in \mathbb N$ such that $P=I:f$ for some $f\in R_c$. A classical result of Brodmann asserts that $Ass(R/I^n)$ is constant for large $n$. So it makes sense to consider a prime ideal $P\in Ass(R/I^n)$ for all the large $n$ and investigate how $v_P(I^n)$ depends on $n$. We prove that $v_P(I^n)$ 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.