On the vanishing of local homology modules

TL;DR

This paper investigates the conditions under which local homology modules vanish, proposing a dual to Grothendieck's Vanishing Theorem and proving it in specific cases.

Contribution

It introduces a conjecture about the vanishing of local homology modules beyond a certain degree and proves this in several cases, extending existing theoretical frameworks.

Findings

Conjecture that $H^{a}_i(M)=0$ for all $i> ext{Mag}_R M$

Proved the conjecture in several specific cases

Provides a dual perspective to Grothendieck's Vanishing Theorem

Abstract

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $M$ an $R$-module. We intend to establish the dual of Grothendieck's Vanishing Theorem for local homology modules. We conjecture that $H^{\fa}_i(M)=0$ for all $i>\Mag_RM$. We prove this in several cases.