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

Dipankar Ghosh, Siddhartha Pramanik

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 $(0:_{U}I) = 0$.

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

Abstract

Let be a Noetherian -graded ring. Let , and be finitely generated graded -modules with . For a homogeneous ideal , and for each fixed , we show the asymptotic linearity of v-numbers of the graded modules and as functions of . Moreover, under some conditions on and respectively, we prove similar behaviour for v-numbers of and . The last result is obtained by proving the asymptotic linearity of v-number of , where , and are graded submodules of a finitely generated graded -module such that and .
Paper Structure (4 sections, 11 theorems, 32 equations)

This paper contains 4 sections, 11 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Asymptotic v-numbers of $\mathop{\mathrm{Ext}}\nolimits_R^k(L,I^nM/I^nN)$ and $\mathop{\mathrm{Tor}}\nolimits_k^R(L,I^nM/I^nN)$
  3. Asymptotic v-numbers of graded (co)homology modules
  4. Some examples

Key Result

Theorem 1.3

With setup, fix $k \in \mathbb{N}$. Set $H := \mathop{\mathrm{Ext}}\nolimits_R^k(L,M)$ and $H_n := \mathop{\mathrm{Ext}}\nolimits_R^k(L,M/I^n M)$, or $H := \mathop{\mathrm{Tor}}\nolimits_k^R(L,M)$ and $H_n := \mathop{\mathrm{Tor}}\nolimits_k^R(L,M/ I^n M)$ for all $n\geqslant 0$. Assume that $(0 :_{

Theorems & Definitions (29)

  • Definition 1.2
  • Theorem 1.3: See Theorems \ref{['thm:main-Ext']} and \ref{['thm:main-Tor']} for stronger results
  • Remark 1.4
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • Remark 3.2
  • ...and 19 more