Lemma 59.51.2. Let $I$ be a directed set. Let $(X_ i, f_{i'i})$ be an inverse system of schemes over $I$ with affine transition morphisms. Let $X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i$. With notation as in Topologies, Lemma 34.4.12 we have

$X_{affine, {\acute{e}tale}} = \mathop{\mathrm{colim}}\nolimits (X_ i)_{affine, {\acute{e}tale}}$

as sites in the sense of Sites, Lemma 7.18.2.

Proof. Let us first prove this when $X$ and $X_ i$ are quasi-compact and quasi-separated for all $i$ (as this is true in all cases of interest). In this case any object of $X_{affine, {\acute{e}tale}}$, resp. $(X_ i)_{affine, {\acute{e}tale}}$ is of finite presentation over $X$. Moreover, the category of schemes of finite presentation over $X$ is the colimit of the categories of schemes of finite presentation over $X_ i$, see Limits, Lemma 32.10.1. The same holds for the subcategories of affine objects étale over $X$ by Limits, Lemmas 32.4.13 and 32.8.10. Finally, if $\{ U^ j \to U\}$ is a covering of $X_{affine, {\acute{e}tale}}$ and if $U_ i^ j \to U_ i$ is morphism of affine schemes étale over $X_ i$ whose base change to $X$ is $U^ j \to U$, then we see that the base change of $\{ U^ j_ i \to U_ i\}$ to some $X_{i'}$ is a covering for $i'$ large enough, see Limits, Lemma 32.8.15.

In the general case, let $U$ be an object of $X_{affine, {\acute{e}tale}}$. Then $U \to X$ is étale and separated (as $U$ is separated) but in general not quasi-compact. Still, $U \to X$ is locally of finite presentation and hence by Limits, Lemma 32.10.5 there exists an $i$, a quasi-compact and quasi-separated scheme $U_ i$, and a morphism $U_ i \to X_ i$ which is locally of finite presentation whose base change to $X$ is $U \to X$. Then $U = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} U_{i'}$ where $U_{i'} = U_ i \times _{X_ i} X_{i'}$. After increasing $i$ we may assume $U_ i$ is affine, see Limits, Lemma 32.4.13. To check that $U_ i \to X_ i$ is étale for $i$ sufficiently large, choose a finite affine open covering $U_ i = U_{i, 1} \cup \ldots \cup U_{i, m}$ such that $U_{i, j} \to U_ i \to X_ i$ maps into an affine open $W_{i, j} \subset X_ i$. Then we can apply Limits, Lemma 32.8.10 to see that $U_{i, j} \to W_{i, j}$ is étale after possibly increasing $i$. In this way we see that the functor $\mathop{\mathrm{colim}}\nolimits (X_ i)_{affine, {\acute{e}tale}} \to X_{affine, {\acute{e}tale}}$ is essentially surjective. Fully faithfulness follows directly from the already used Limits, Lemma 32.10.5. The statement on coverings is proved in exactly the same manner as done in the first paragraph of the proof. $\square$

Comment #5576 by on

It seems to me that $\mathop{\mathrm{colim}}\nolimits (X_ i)_{\mathrm{\acute{e}t}}$ is not considered in situation 7.18.1. Given morphisms for the corresponding continuous functors of sites, we have in general because $u_a \circ u_b$ sends an etale $X_i$-scheme $U$ to $X_k\times_{X_i} U$, while $u_c$ sends $U$ to $X_k \times_{X_j} X_j \times_{X_i} U$.

There is a unique isomorphism $u_c \to u_a \circ u_b$ but it is not the identity. So we are not in the situation 7.18.1. I think we need to apply 4.36.4 to pass to that situation.

Comment #5756 by on

Of course you are correct and we are being sloppy here. Of course, almost everybody is sloppy in this sense: almost everybody identifies a triple fibre product $X \times_S Y \times_T Z$ with both $(X \times_S Y) \times_T Z$ and with $X \times_S (Y \times_T Z)$ and almost everybody identifies $X \times_S S$ with $X$.

My feeling is that in order to get rid of the sloppyness we should upgrade the discussion in Section 7.18 rather than change what is written in this proof. As you indicate in your comment we should extend Situation 7.18.1 to the case where we have a fibred category over $\mathcal{I}$ and the functors $u_a$ are the pullback functors and the fibre categories are endowed with the structure of a site such that the pullback functors are continuous and define morphisms of sites. This should first be discussed without any reference to 7.18.1 of course (because it isn't general enough otherwise). There should also be a discussion of a "dual" notion where we have a cofibred category whose fibre categories are endowed with the structure of a site and whose pushforward functors are cocontinuous. This case is very different and not amenable to taking the colimit of sites but can sometimes be useful.

This should then all be related to the material in Sections 84.3 and 21.38.

I've put it on a long todo list I have somewhere.

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