A connectedness result in positive characteristic

2006/03/09 by Anurag K. Singh, Uli Walther, Singh, Anurag K. +1
Mathematics · #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC

paper · pdf · doi:10.48550/arxiv.math/0603234

arxiv created 2006/03/09 · arxiv updated 2009/12/01

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 H¹_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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