A Note on Associated Primes and Bockstein Homomorphisms of Local Cohomology Modules for Ramified Regular Local Rings

TL;DR

This paper investigates how properties of associated primes of local cohomology modules transfer from unramified to ramified regular local rings, extending results on Bockstein homomorphisms and associated prime finiteness.

Contribution

It demonstrates the transfer of property P from unramified to ramified regular local rings and extends Bockstein homomorphism results to ramified cases.

Findings

Associated primes of local cohomology modules are finite when they do not contain p.

Property P can descend from unramified to ramified regular local rings.

Bockstein homomorphism vanishes under certain conditions in ramified rings.

Abstract

For a Noetherian regular ring $S$ and for a fixed ideal $J\subset S$, assume that the associated primes of local cohomology module $H^i_J(S)$ does not contain $p$ for some $i\geq 0$, and we call this as a property $\textit{\textbf{P}}^{i,p}_{J,S}$ or $\textit{\textbf{P}}$ for brevity. Recently, in Theorem 1.2 of \cite{Nu1}, it is proved that in a Noetherian regular local ring $S$ and for a fixed ideal $J\subset S$, associated primes of local cohomology module $H^i_J(S)$ for $i\geq 0$ is finite, if it does not contain $p$. In this paper, we study how the property $\textit{\textbf{P}}$ (as mentioned above) can come down from unramified regular ring to ramified regular local ring. In \cite{SW}, Bockstein homomorphism is studied in the context to the finiteness of associated primes of local cohomology modules for the ring of integers. There it is shown that if $p$ is nonzero divisor of Koszul homology then Bockstein homomorphism is a zero map (see, Theorem 3.1 of \cite{SW}). Here, in this paper, as a consequence of property $\textit{\textbf{P}}$, we extend the result of Theorem 3.1 of \cite{SW} to the ramified regular local ring.