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

Tony J. Puthenpurakal

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 $i$.

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

Abstract

Let be a regular ring of dimension containing a field of characteristic zero. If is an -module let . Let be a prime ideal in of height . We show that if satisfies Serre's condition then is a finite set. As an application of our techniques we prove that if is a prime ideal in such that is regular for any non-maximal prime ideal then has finitely many associate primes for all .

Paper Structure

This paper contains 4 sections, 7 theorems, 8 equations.

Table of Contents

  1. introduction
  2. Passage to a ring containing an uncountable field
  3. Construction
  4. Proof of Theorems \ref{['main']} and \ref{['main-r1']}

Key Result

Theorem 1.2

Let $R$ be a regular ring of dimension $d$, containing a field of characteristic zero. Let $P$ be 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 $i$.

Theorems & Definitions (12)

  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 3.2
  • Proposition 3.3
  • Theorem 3.5
  • proof
  • ...and 2 more