Bockstein homomorphisms in local cohomology

TL;DR

This paper proves that Bockstein homomorphisms in local cohomology vanish for almost all primes, supporting the conjecture that local cohomology modules over polynomial rings have finitely many associated primes.

Contribution

It demonstrates that Bockstein homomorphisms are zero for all but finitely many primes, providing evidence for Lyubeznik's conjecture on associated primes.

Findings

Bockstein homomorphisms vanish for all but finitely many primes.

Supports Lyubeznik's conjecture on finiteness of associated primes.

Provides a new approach to understanding local cohomology in algebraic geometry.

Abstract

Let $R$ be a polynomial ring in finitely many variables over the integers, and fix an ideal $I$ of $R$. We prove that for all but finitely prime integers $p$, the Bockstein homomorphisms on local cohomology, $H^k_I(R/pR)\to H^{k+1}_I(R/pR)$, are zero. This provides strong evidence for Lyubeznik's conjecture which states that the modules $H^k_I(R)$ have a finite number of associated prime ideals.