Non-finite type \'etale sites over fields

TL;DR

This paper investigates the finite type-ness of the étale site over fields, proposing a conjecture linking it to the field's finite extensions and cohomological dimension, and proves it in specific cases.

Contribution

It introduces a conjecture relating finite type-ness of étale sites over fields to their cohomological properties and proves it for certain classes of fields.

Findings

Conjecture: étale site of Sm/k is of finite type iff k has a finite extension with finite cohomological dimension.

Proved the conjecture for countable fields.

Established the conjecture when the p-cohomological dimension is infinite for infinitely many primes p.

Abstract

We consider the notion of finite type-ness of a site introduced by Morel and Voevodsky, for the \'etale site of a field. For a given field $k$, we conjecture that the \'etale site of $Sm/k$ is of finite type if and only if the field $k$ admits a finite extension of finite cohomological dimension. We prove this conjecture in some cases, e.g. in the case when $k$ is countable, or in the case when the $p$-cohomological dimension $cd_p(k)$ is infinite for infinitely many primes $p$.