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

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 ${\mathrm{Ext}}_R^k(L,I^{n}M/I^{n}N)$ and ${\mathrm{Tor}}_k^R(L,I^{n}M/I^{n}N)$ as functions of $n$. Moreover, under some conditions on ${\mathrm{Ext}}_R^k(L,M)$ and ${\mathrm{Tor}}_k^R(L,M)$ respectively, we prove similar behaviour for v-numbers of ${\mathrm{Ext}}_R^k(L,M/I^{n}N)$ and ${\mathrm{Tor}}_k^R(L,M/I^{n}N)$. 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 $(0{:}_{U}I)=0$.

11 pages, Comments and suggestions are welcome

References in corpus (1)

• On the asymptotic behaviour of the Vasconcelos invariant for graded modules