A note on the associated primes of local cohomology modules for regular local rings

Abstract

Let $R$ be a regular local ring. In this note, we prove that $Ass_RH^2_I(R)$ is finite for any ideal $I$ of $R$. We also give a sufficient condition for $Ass_RH^3_{(x,y,z)}(R)$ to be finite for $x, y$ an $R$-regular sequence and $z\in R$, which would imply that Lyubeznik's conjecture is true in the regular local rings case.