5

The etale fundamental group, as explained in SGA 1 Expose 5 and various other notes I've read, always makes the assumption that the scheme $S$ (for which one intends to construct a fundamental group), is locally noetherian.

How necessary is this assumption?

For example, let $\text{FEt}_S$ be the category of schemes finite etale over a fixed connected scheme $S$. Is $\text{FEt}_S$ a galois category when $S$ isn't locally noetherian?

From the viewpoint of galois categories, it seems that almost all the axioms for a galois category are obvious for $\text{FEt}_S$ (without any noetherian hypotheses), except possibly the conditions:

  1. For any $X\in\text{FEt}_S$ and a finite group $G$ acting on $X$ by automorphisms over $S$, the quotient $X/G$ exists in $\text{FEt}_S$.

  2. For a geometric point $s\in S$, and any $X\in\text{FEt}_S$ acted on by a finite group $G$ of $S$-automorphisms, the fiber functor $F_s$ satisfies $F(X)/G\cong F(X/G)$

The latter appears to follow from SGA 1 Expose 5 beginning of section 2 (as presented here http://arxiv.org/abs/math/0206203)

However, I'm not sure if the first is true. Since $X\rightarrow S$ is finite hence affine, you can reduce to the case where $X,S$ are both affine. In this case if $X = \text{Spec }A$, then $X/G$ is just $\text{Spec }A^G$, but I don't know if $\text{Spec }A^G$ is finite etale over $S$.

oxeimon
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    That $\mathrm{FET}_S$ is a Galois category holds for all $S$. You can find the proof in Lenstra's notes on galois theory. However, deeper theorems about $\pi_1$ require finiteness assumptions. – Martin Brandenburg Jul 24 '14 at 15:16
  • @MartinBrandenburg Can you give some examples of these deeper theorems that depend on noetherian hypotheses? – oxeimon Jul 24 '14 at 16:21
  • There are a lot of theorems for which I'm not positive the Noetherian hypotheses are necessary, but the only proofs I know make use of the hypothesis. For exmample, the fact that a map between two finite etale covers is determined by its action on a geometric point. – Alex Youcis Jul 24 '14 at 16:38
  • @AlexYoucis Isn't that just one of the many consequences of FET_S being a galois category? – oxeimon Jul 24 '14 at 17:06
  • @oxeimon I'm not entirely sure. I forget what the axioms laid out in Lenstra are precisely. I was just thinking about Lei Fu's etale cohomology book, and what he required. The Noetherian hypotheses often times came in at odd points to prove some random lemma like the one I just stated. It seems weird that Fu assumes Noetherian hypotheses he doesn't need, since he usually just does 'passage to the limit' to obtain results over all rings. – Alex Youcis Jul 24 '14 at 17:16

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