Examples of local Cohomology Modules for Ramified Regular Local Rings having Finite Set of Associated Primes

TL;DR

This paper constructs examples of local cohomology modules over ramified regular local rings with a finite set of associated primes, addressing an open case of Lyubeznik's conjecture.

Contribution

It provides explicit examples of local cohomology modules with finitely many associated primes in ramified regular local rings, extending known results.

Findings

Examples of local cohomology modules with finite associated primes

Application of recent theorem to ramified regular local rings

Addresses open case of Lyubeznik's conjecture

Abstract

Lyubeznik's conjecture, (\cite{Ly1}, Remark 3.7) asserts the finiteness of the set ssociated primes of local cohomology modules for regular rings. But, in the case of ramified regular local ring, it is open. Recently, in Theorem 1.2 of \cite{Nu1}, it is proved that in any Noetherian regular local ring $S$ and for a fixed ideal $J\subset S$, associated primes of local cohomology $H^i_J(S)$ for $i\geq 0$ is finite, if it does not contain $p$. In this paper, we use this result to construct examples of local cohomology modules for ramified regular local ring so that they have finitely many associated primes.