Local cohomology of ideals and the $R_n$ condition of Serre

Abstract: Let $R$ be a regular ring of dimension $d$ containing a field $K$ of characteristic zero. If $E$ is an $R$-module let $Ass^i E = { Q \in \ Ass E \mid \ height Q = i }$. Let $P$ be a prime ideal in $R$ of height $g$. We show that if $R/P$ satisfies Serre's condition $R_i$ then $Ass^{g+i+1}H^{g+1}_P(R)$ is a finite set. As an application of our techniques we prove that if $P$ is a prime ideal in $R$ such that $(R/P)_\mathfrak{q}$ is regular for any non-maximal prime ideal $\mathfrak{q}$ then $H^i_P(R)$ has finitely many associate primes for all $i$.