A connectedness result in positive characteristic

Abstract:

Let $(R,\mathfrak {m})$ be a complete local ring of dimension at least two, which contains a separably closed coefficient field of positive characteristic. Using a vanishing theorem of Peskine-Szpiro, Lyubeznik proved that the local cohomology module $H^1_{\mathfrak {m}}(R)$ is Frobenius-torsion if and only if the punctured spectrum of $R$ is connected in the Zariski topology. We give a simple proof of this theorem and, more generally, a formula for the number of connected components in terms of the Frobenius action on $H^1_{\mathfrak {m}}(R)$.