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arXiv · 2405.06612

Non-finite type étale sites over fields

Sujeet Dhamore·Amit Hogadi·Rakesh Pawar
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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 $p$.

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Sujeet Dhamore, Amit Hogadi, Rakesh Pawar. 2025-04-16. Non-finite type étale sites over fields. https://doi.org/10.1016/j.jalgebra.2024.12.036

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