Bockstein homomorphisms in local cohomology

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, $H_I^k(R/pR)\to H_I^{k+1}(R/pR)$, are zero. This provides strong evidence for Lyubeznik's conjecture which states that the modules $H_I^k(R)$ have a finite number of associated prime ideals.