Asymptotic v-numbers of graded (co)homology modules involving powers of an ideal

TL;DR

This paper investigates the asymptotic linearity of v-numbers associated with graded (co)homology modules involving powers of an ideal over a Noetherian graded ring, revealing new linearity properties under certain conditions.

Contribution

It establishes the asymptotic linearity of v-numbers for Ext and Tor modules involving powers of an ideal, extending understanding of their long-term behavior.

Findings

Proves asymptotic linearity of v-numbers for Ext modules.

Shows similar linearity for Tor modules under conditions.

Extends results to modules involving sums and powers of ideals.

Abstract

Let $R$ be a Noetherian $\mathbb{N}$-graded ring. Let $L$, $M$ and $N$ be finitely generated graded $R$-modules with $N \subseteq M$. For a homogeneous ideal $I$, and for each fixed $k \in \mathbb{N}$, we show the asymptotic linearity of v-numbers of the graded modules $ {\rm Ext}_R^{k}(L,{I^{n}M}/{I^{n}N})$ and ${\rm Tor}_k^{R}(L,{I^{n}M}/{I^{n}N})$ as functions of $n$. Moreover, under some conditions on ${\rm Ext}_R^k(L,M)$ and ${\rm Tor}_k^R(L,M)$ respectively, we prove similar behaviour for v-numbers of ${\rm Ext}_R^{k}(L,{M}/{I^{n}N})$ and $ {\rm Tor}_k^{R}(L,{M}/{I^{n}N})$. The last result is obtained by proving the asymptotic linearity of v-number of $(U+I^{n}V)/I^{n}W$, where $U$, $V$ and $W$ are graded submodules of a finitely generated graded $R$-module such that $W \subseteq V$ and $(0:_{U}I) = 0$.