\'Etale cohomological dimension, a conjecture of Lyubeznik and bounds for arithmetic rank

TL;DR

This paper establishes criteria for bounding étale cohomological dimension in projective spaces, provides counterexamples to Lyubeznik's conjecture, and relates algebraic invariants to geometric properties of schemes.

Contribution

It introduces a new criterion for étale cohomological dimension bounds, constructs examples with smaller dimension than conjectured, and links algebraic invariants to geometric cohomological properties.

Findings

A criterion for étale cohomological dimension in projective space.

Counterexamples to Lyubeznik's conjecture with smaller cohomological dimension.

Upper bounds for arithmetic rank based on algebraic invariants.

Abstract

We produce a criterion for open sets in projective $n$-space over a separably closed field to have \'etale cohomological dimension bounded by $2n-3$. We use the criterion to exhibit a scheme for which \'etale 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 \'etale 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.