On Rees localiti es and $H_i$-local rings

Abstract

The main theorem gives a necessary and sufficient condition for each Rees locality $\mathscr{L}=R[tb, u]_{(M,tb,u)}$ of a local ring $(R, M) $ with respect to a principal ideal $bR$ in $R$ to be either an $H_{i}$-ring (that is, for all prime ideals $p$ in $\mathscr{L}$ such that height $p=i$, depth $p=$ altitude $\mathscr{L}-i$) or a homogeneously $H_{i}$-ring (same condition holds for homogeneous $p$). Numerous corollaries follow concerning the cases: $R$ is complete; $R$ is Henselian; and, $\mathscr{L}$ is $H_i$, for all $i\geqq 0$. A generalization to ideals generated by more than one element is given, and we relate the results to two of the chain conjectures on prime ideals.