On Finite Type Property of Étale Sites

Abstract

In this thesis, we consider the notion of finite type-ness or Postnikov completeness of a site introduced by Morel and Voevodsky. One of the important consequences of having a finite type site is the existence of an exact fibrant resolution functor which preserves fibrations. The finite type-ness of the Nisnevich site has crucial consequences in the development of A^1-homotopy theory, in particular in obstruction theory. With the motivation towards the development of étale A1-homotopy theory, we investigate the finite type-ness of the étale site (Sm/k)_ét of finite type smooth schemes over a field k. We conjecture that this étale site is of finite type if and only if k admits a finite extension L with finite cohomological dimension. Our main result proves this conjecture when the absolute Galois group G_k is first-countable, which holds, in particular, for countable fields. Additionally, we establish necessary conditions for the finite type-ness of this site by proving that if k has arbitrarily large order higher degree cohomologies, which includes the case when cd_p(k) is infinite for infinitely many primes, then this site is not of finite type.