Associated primes of Local cohomology modules over Regular rings

TL;DR

This paper proves the finiteness of associated primes of a specific local cohomology module over regular rings of characteristic zero and explores the topological connectedness of certain spectra related to these ideals.

Contribution

It establishes the finiteness of associated primes of the local cohomology module $H^{d-1}_I(R)$ over regular rings and analyzes the connectedness of spectra in relation to ideal height.

Findings

Finiteness of $Ass ext{ } H^{d-1}_I(R)$ for regular rings of characteristic zero.

Connectedness of $Spec^ extcircled{ }(R_P/IR_P)$ for primes with certain height conditions.

Applicable to ideals with uniform height across minimal primes.

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 $Ass \ H^{d-1}_I(R)$ is a finite set. As an application we show that if $I$ is an ideal of height $g$ with $height \ Q = g$ for all minimal primes of $I$ then for all but finitely many primes $P \supseteq I$ with $height \ P \geq g +2$, the topological space $Spec^\circ(R_P/IR_P)$ is connected.