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