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

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.

12pp

References in corpus (1)

• On the Arithmetical Rank of Certain Segre Embeddings