A connectedness result in positive characteristic
Anurag K. Singh, Uli Walther
Abstract
Let (R,m) be a complete local ring of positive dimension, which contains a separably closed coefficient field of prime characteristic. Using a vanishing theorem of Peskine-Szpiro, Lyubeznik proved that every element of the local cohomology module H1m(R) is killed by an iteration of the Frobenius map if and only if R has dimension at least two and its punctured spectrum is connected in the Zariski topology. We give a simple proof of this theorem and of a variation which, more generally, yields the number of connected components.
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