Non-finite type étale sites over fields

TL;DR

The paper investigates when the étale site $(Sm/k)_{\acute{e}t}$ is of finite type in the Morel–Voevodsky sense, proposing that this occurs exactly when some finite extension $L/k$ has finite cohomological dimension $cd(L)$. It develops a constructible obstruction framework based on Postnikov truncations and Eilenberg–MacLane objects, reducing finite-type questions to Galois-cohomological data. The main contributions prove the conjecture for countable fields and establish sharp non-finite-type criteria under unbounded cohomological dimensions, with a key Lemma ftcondition driving the argument. An Appendix ties étale-$\mathbb{A}^1$-connectedness to rational connectedness in characteristic zero, highlighting the interplay between étale homotopy theory and birational geometry, and suggesting broader implications for étale $\mathbb{A}^1$-homotopy theory on 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 $Sm/k$ 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 $p$.

Theorems & Definitions (17)

• Definition 1.1
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Proposition 1.7
• Definition 2.1
• Example 3.1
• Lemma 3.2