Associated primes of local cohomology modules over regular rings

Tony J. Puthenpurakal

Vol. 282 (2016), No. 1, 233–255

DOI: 10.2140/pjm.2016.282.233

Abstract

Let $R$ be an excellent regular ring of dimension $d$ containing a field $K$ of characteristic zero. Let $I$ be an ideal in $R$. We show that $Ass<!--nolimits-->H_I^{d-1}\left(R\right)$ is a finite set. As an application, we show that if $I$ is an ideal of height $g$ with $height<!--nolimits-->Q=g$ for all minimal primes of $I$ then for all but finitely many primes $P\supseteq I$ with $height<!--nolimits-->P\ge g+2$, the topological space ${Spec<!--nolimits-->}^{\circ }\left(R_P∕I{R}_P\right)$ is connected. We also show that to prove a conjecture of Lyubeznik (regarding finiteness of associate primes for local cohomology modules) for all excellent regular rings of dimension $\le d$ containing a field of characteristic zero, it suffices to prove ${Ass<!--nolimits-->}_S{H}_J^{g+1}\left(S\right)$ is finite for all ideals $J$ in $S$ of height $g$ (here $0\le g\le d$), where $S$ is an excellent regular domain of dimension $\le d$ containing an uncountable field of characteristic zero.

Keywords

local cohomology, associate primes, $D$-modules

Mathematical Subject Classification 2010

Primary: 13D45

Secondary: 13D02, 13H10

Milestones

Received: 10 June 2015

Revised: 20 August 2015

Accepted: 22 August 2015

Published: 24 February 2016