A connectedness result in positive characteristic

Details A connectedness result in positive characteristic A connectedness result in positive characteristic

2006-03-09

arxiv.org/abs/math/0603234v1

Mathematics

Commutative Algebra

Scientific paper

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 $H^1_m(R)$ 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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