In certain concrete situations, I can show that the small étale topos of an inverse limit of schemes is the inverse limit of the associated toposes, for example, if $X$ is a (qcqs) scheme relative over some affine scheme $\operatorname{Spec}(R)$, and $S$ is a filtered colimit of $R$-algebras where $S=\varinjlim S_i$, I can show that there is an equivalence of small étale toposes

$$\operatorname{\acute{E}t}(X\times_{\operatorname{Spec}(R)} \operatorname{Spec}(S))\simeq \varprojlim \operatorname{\acute{E}t}(X\times_{\operatorname{Spec}(R)} \operatorname{Spec}(S_i)).$$

(Note: The inverse limit here is being taken in the category of toposes, not in the category of locally ringed toposes!!)

I heard from a friend that there is a general statement that given an inverse system $\{X_i\}$ of schemes with affine transition maps and limit $X$, then we have an equivalence of étale toposes:

$$\operatorname{\acute{E}t}(X)\simeq \varprojlim\operatorname{\acute{E}t}(X_i).$$

I looked high and low for a reference, and I couldn't find a proof of this statement. Can someone provide a reference?

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    $\begingroup$ The first "asking for a friend" MO question? :-) $\endgroup$ – LSpice Oct 17 '19 at 21:32
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    $\begingroup$ I'm not entirely sure how the 2-limit of topoi is constructed, but is it possible that there are some finiteness issues (even in your first situation)? For example, I think that the sheaf of sets $\coprod_{d > 0} \underline{\mathbf Z/2}_{V(x-\sqrt{d}y)}$ on $\mathbf A^2_{\bar{\mathbf Q}}$ (the disjoint union of the constant $\mathbf Z/2$-sheaf supported on $V(x-\sqrt{d}y) \subseteq \mathbf A^2_{\bar{\mathbf Q}}$ for all $d$) does not descend to any finite extension $\mathbf Q \subseteq K$. I'm not sure if this is a problem... $\endgroup$ – R. van Dobben de Bruyn Oct 18 '19 at 3:55
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    $\begingroup$ @R. van Dobben de Bruyn It's definitely true for affine schemes by an argument using the sites. I don't understand the definition of your sheaf, but any étale S-algebra descends to some S_i, so you get that in the category of sites, the étale site of S is the colimit of the S_i. Standard arguments then give the statement for the toposes. $\endgroup$ – Harry Gindi Oct 18 '19 at 8:19
  • $\begingroup$ I agree about the site-theoretic statement. I just don't see how that says anything about sheaves. But again, I don't know what the 2-limit of topoi is (but it seems I was thinking more about a 2-colimit). $\endgroup$ – R. van Dobben de Bruyn Oct 18 '19 at 14:09
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    $\begingroup$ See the references in the proof of Lemma 1.17, Chept III, of Milne Etale Cohomology. $\endgroup$ – anon Oct 19 '19 at 14:04

I think I found the answer (with some help) in SGA 4, Exposé VII, Lemma 5.6.

The statement (translated and with context added) reads:

Lemma 5.6 [Given a cofiltered poset $I$ and a functor $\mathscr{X}:I\to \mathbf{Sch}$ with affine transition maps and limit $X$, valued in qcqs schemes], the induced functor $$\varinjlim_{\mathcal{G}_{\mathscr{X}}/I} \mathcal{G}_{\mathscr{X}}\to \mathcal{G}_X$$ determines an equivalence, respecting the topologies, of the site given by the inductive limit of the restricted étale sites of the $\mathscr{X}_i$ and the restricted étale site of $X$.

(where the restricted étale site $\mathcal{G}_S$ is full subcategory of the étale site spanned by the objects of finite presentation). This implies an equivalence of topoi by SGA4, Exposé VII, Corollary 3.2, since the schemes are quasi-separated.

I'm not sure if it's true without assuming that the schemes appearing in the limit are qcqs, but I'd like to know!

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Milne Etale Cohomology, III Lemma 1.16:

Let $I$ be a small filtered category and $i\rightsquigarrow X_{i}$ a contravariant functor from $I$ to schemes over $X$. Assume that all schemes $X_{i}$ are quasi-compact and that the maps $X_{i}\leftarrow X_{j}$ are affine. Let $X_{\infty}=\varprojlim X_{i}$, and, for a sheaf $F$ on $X_{\mathrm{et}}$, let $F_{i}$ and $F_{\in fty}$ be its inverse images on $X_{i}$ and $X_{\infty}$ respectively. Then $$ \injlim H^{p}((X_{i})_{\mathrm{et}},F_{i})\overset{\simeq}{\longrightarrow}H^{p}((X_{\infty})_{\mathrm{et}},F_{\infty}). $$

The proof is highly technical. It is based on the fact [EGA. IV.17] that the category of etale schemes of finite-type over $X_{\infty}$ is the direct limit of the categories of such schemes over the $X_{i}$ and (III, 3) that etale cohomology commutes with direct limits of sheaves. (See [SGA. 4, VII, 5.8] or Artin, Grothendieck Topologies, Harvard notes, 1962, III, 3] for the details.)

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  • $\begingroup$ I tracked back all of these results, and it looks like you need quasi-separated as well. The distinction might come from Milne requiring his schemes to be separated by definition, maybe? The proof in SGA also uses the quasi-separatedness to say that you can replace the small étale site with its finite presentation variant. $\endgroup$ – Harry Gindi Oct 22 '19 at 21:06

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