Surjectivity of some local cohomology map and the second vanishing theorem

TL;DR

This paper investigates the second vanishing theorem in local cohomology, providing partial results for ramified regular local rings by exploring surjectivity properties of local cohomology maps.

Contribution

It extends the second vanishing theorem to the ramified case using surjective elements, a novel approach in local cohomology theory.

Findings

Partial proof of the second vanishing theorem for ramified regular local rings

Introduction of surjective elements as a key tool in local cohomology

Connections established between topological properties and local cohomology in ramified settings

Abstract

The second vanishing theorem has a long history in the theory of local cohomology modules, which connects the vanishing of a complete regular local ring with a topological property of the punctured spectrum of the ring under some conditions. However, the case of complete ramified regular local rings is unresolved. In this paper, we give a partial answer to the second vanishing theorem in the ramified case. Our proof is inspired by the theory of surjective elements in the theory of local cohomology.