An elementary proof of Grothendieck's Non-vanishing Theorem

TL;DR

This paper provides a straightforward, elementary proof of Grothendieck's Non-vanishing Theorem, establishing that certain local cohomology modules are non-zero for finitely generated modules over Noetherian local rings.

Contribution

It introduces a simplified proof of Grothendieck's Non-vanishing Theorem, making the result more accessible and easier to understand.

Findings

Proof confirms non-vanishing of local cohomology modules in the specified setting.

Simplifies the understanding of Grothendieck's Non-vanishing Theorem.

Enhances accessibility of key algebraic geometry concepts.

Abstract

We give an elementary proof of Grothendieck's non-vanishing Theorem: For a finitely generated non-zero module $M$ over a Noetherian local ring $A$ with maximal ideal $\m$, the local cohomology module $H^{\dim M}_{\m}(M)$ is non-zero.