Étale cohomological dimension, a conjecture of Lyubeznik and bounds for arithmetic rank

2010/11/30 by Manoj Kummini, Uli Walther, Kummini, Manoj +1
Mathematics · #13E15 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 14F20 #Secondary 13D02 #math.AC #math.AG #msc:13D02 #msc:13E15 #msc:14F20

paper · pdf · doi:10.48550/arxiv.1011.6648

12pp

arxiv created 2010/11/30 · arxiv updated 2010/12/01

Abstract

We produce a criterion for open sets in projective n-space over a separably closed field to have étale cohomological dimension bounded by 2n-3. We use the criterion to exhibit a scheme for which étale cohomological dimension is smaller than what a conjecture of G.~Lyubeznik predicts; the discrepancy is of arithmetic nature. For a monomial ideal, we relate extremal graded Betti numbers and étale cohomological dimension of the complement of the corresponding subspace arrangement. Moreover, we derive upper bounds for its arithmetic rank in terms of invariants distilled from the lcm-lattice.

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