Non-finite type étale sites over fields

Abstract

We consider the notion of finite type-ness of a site introduced by Morel and Voevodsky, for the étale site of a field. For a given field k, we conjecture that the étale site of 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 is infinite for infinitely many primes p.