Definition 61.12.1. Let $T$ be a scheme. A pro-étale covering of $T$ is a family of morphisms $\{ f_ i : T_ i \to T\} _{i \in I}$ of schemes such that each $f_ i$ is weakly-étale and such that for every affine open $U \subset T$ there exists $n \geq 0$, a map $a : \{ 1, \ldots , n\} \to I$ and affine opens $V_ j \subset T_{a(j)}$, $j = 1, \ldots , n$ with $\bigcup _{j = 1}^ n f_{a(j)}(V_ j) = U$.
61.12 The pro-étale site
In this section we only discuss the actual definition and construction of the various pro-étale sites and the morphisms between them. The existence of weakly contractible objects will be done in Section 61.13.
The pro-étale topology is a bit like the fpqc topology (see Topologies, Section 34.9) in that the topos of sheaves on the small pro-étale site of a scheme depends on the choice of the underlying category of schemes. Thus we cannot speak of the pro-étale topos of a scheme. However, it will be true that the cohomology groups of a sheaf are unchanged if we enlarge our underlying category of schemes, see Section 61.31.
We will define pro-étale coverings using weakly étale morphisms of schemes, see More on Morphisms, Section 37.64. The reason is that, on the one hand, it is somewhat awkward to define the notion of a pro-étale morphism of schemes, and on the other, Proposition 61.9.1 assures us that we obtain the same sheaves1 with the definition that follows.
To be sure this condition implies that $T = \bigcup f_ i(T_ i)$. Here is a lemma that will allow us to recognize pro-étale coverings. It will also allow us to reduce many lemmas about pro-étale coverings to the corresponding results for fpqc coverings.
Lemma 61.12.2. Let $T$ be a scheme. Let $\{ f_ i : T_ i \to T\} _{i \in I}$ be a family of morphisms of schemes with target $T$. The following are equivalent
$\{ f_ i : T_ i \to T\} _{i \in I}$ is a pro-étale covering,
each $f_ i$ is weakly étale and $\{ f_ i : T_ i \to T\} _{i \in I}$ is an fpqc covering,
each $f_ i$ is weakly étale and for every affine open $U \subset T$ there exist quasi-compact opens $U_ i \subset T_ i$ which are almost all empty, such that $U = \bigcup f_ i(U_ i)$,
each $f_ i$ is weakly étale and there exists an affine open covering $T = \bigcup _{\alpha \in A} U_\alpha $ and for each $\alpha \in A$ there exist $i_{\alpha , 1}, \ldots , i_{\alpha , n(\alpha )} \in I$ and quasi-compact opens $U_{\alpha , j} \subset T_{i_{\alpha , j}}$ such that $U_\alpha = \bigcup _{j = 1, \ldots , n(\alpha )} f_{i_{\alpha , j}}(U_{\alpha , j})$.
If $T$ is quasi-separated, these are also equivalent to
each $f_ i$ is weakly étale, and for every $t \in T$ there exist $i_1, \ldots , i_ n \in I$ and quasi-compact opens $U_ j \subset T_{i_ j}$ such that $\bigcup _{j = 1, \ldots , n} f_{i_ j}(U_ j)$ is a (not necessarily open) neighbourhood of $t$ in $T$.
Proof. The equivalence of (1) and (2) is immediate from the definitions. Hence the lemma follows from Topologies, Lemma 34.9.2. $\square$
Lemma 61.12.3. Any étale covering and any Zariski covering is a pro-étale covering.
Proof. This follows from the corresponding result for fpqc coverings (Topologies, Lemma 34.9.7), Lemma 61.12.2, and the fact that an étale morphism is a weakly étale morphism, see More on Morphisms, Lemma 37.64.9. $\square$
Lemma 61.12.4. Let $T$ be a scheme.
If $T' \to T$ is an isomorphism then $\{ T' \to T\} $ is a pro-étale covering of $T$.
If $\{ T_ i \to T\} _{i\in I}$ is a pro-étale covering and for each $i$ we have a pro-étale covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a pro-étale covering.
If $\{ T_ i \to T\} _{i\in I}$ is a pro-étale covering and $T' \to T$ is a morphism of schemes then $\{ T' \times _ T T_ i \to T'\} _{i\in I}$ is a pro-étale covering.
Proof. This follows from the fact that composition and base changes of weakly étale morphisms are weakly étale (More on Morphisms, Lemmas 37.64.5 and 37.64.6), Lemma 61.12.2, and the corresponding results for fpqc coverings, see Topologies, Lemma 34.9.8. $\square$
Lemma 61.12.5. Let $T$ be an affine scheme. Let $\{ T_ i \to T\} _{i \in I}$ be a pro-étale covering of $T$. Then there exists a pro-étale covering $\{ U_ j \to T\} _{j = 1, \ldots , n}$ which is a refinement of $\{ T_ i \to T\} _{i \in I}$ such that each $U_ j$ is an affine scheme. Moreover, we may choose each $U_ j$ to be open affine in one of the $T_ i$.
Proof. This follows directly from the definition. $\square$
Thus we define the corresponding standard coverings of affines as follows.
Definition 61.12.6. Let $T$ be an affine scheme. A standard pro-étale covering of $T$ is a family $\{ f_ i : T_ i \to T\} _{i = 1, \ldots , n}$ where each $T_ j$ is affine, each $f_ i$ is weakly étale, and $T = \bigcup f_ i(T_ i)$.
We follow the general outline given in Topologies, Section 34.2 for constructing the big pro-étale site we will be working with. However, because we need a bit larger rings to accommodate for the size of certain constructions we modify the constructions slightly.
Definition 61.12.7. A big pro-étale site is any site $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ as in Sites, Definition 7.6.2 constructed as follows:
Choose any set of schemes $S_0$, and any set of pro-étale coverings $\text{Cov}_0$ among these schemes.
Change the function $Bound$ of Sets, Equation (3.9.1.1) into
As underlying category take any category $\mathit{Sch}_\alpha $ constructed as in Sets, Lemma 3.9.2 starting with the set $S_0$ and the function $Bound$.
Choose any set of coverings as in Sets, Lemma 3.11.1 starting with the category $\mathit{Sch}_\alpha $ and the class of pro-étale coverings, and the set $\text{Cov}_0$ chosen above.
\[ Bound(\kappa ) = \max \{ \kappa ^{2^{2^{2^\kappa }}}, \kappa ^{\aleph _0}, \kappa ^+\} . \]
See the remarks following Topologies, Definition 34.3.5 for motivation and explanation regarding the definition of big sites.
It will turn out, see Lemma 61.31.1, that the topology on a big pro-étale site $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ is in some sense induced from the pro-étale topology on the category of all schemes.
Definition 61.12.8. Let $S$ be a scheme. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site containing $S$.
The big pro-étale site of $S$, denoted $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$, is the site $\mathit{Sch}_{pro\text{-}\acute{e}tale}/S$ introduced in Sites, Section 7.25.
The small pro-étale site of $S$, which we denote $S_{pro\text{-}\acute{e}tale}$, is the full subcategory of $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ whose objects are those $U/S$ such that $U \to S$ is weakly étale. A covering of $S_{pro\text{-}\acute{e}tale}$ is any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ with $U \in \mathop{\mathrm{Ob}}\nolimits (S_{pro\text{-}\acute{e}tale})$.
The big affine pro-étale site of $S$, denoted $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$, is the full subcategory of $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ whose objects are affine $U/S$. A covering of $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ is any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ which is a standard pro-étale covering.
It is not completely clear that the small pro-étale site and the big affine pro-étale site are sites. We check this now.
Lemma 61.12.9. Let $S$ be a scheme. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site containing $S$. Both $S_{pro\text{-}\acute{e}tale}$ and $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ are sites.
Proof. Let us show that $S_{pro\text{-}\acute{e}tale}$ is a site. It is a category with a given set of families of morphisms with fixed target. Thus we have to show properties (1), (2) and (3) of Sites, Definition 7.6.2. Since $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is a site, it suffices to prove that given any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ with $U \in \mathop{\mathrm{Ob}}\nolimits (S_{pro\text{-}\acute{e}tale})$ we also have $U_ i \in \mathop{\mathrm{Ob}}\nolimits (S_{pro\text{-}\acute{e}tale})$. This follows from the definitions as the composition of weakly étale morphisms is weakly étale.
To show that $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ is a site, reasoning as above, it suffices to show that the collection of standard pro-étale coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 7.6.2. This follows from Lemma 61.12.2 and the corresponding result for standard fpqc coverings (Topologies, Lemma 34.9.11). $\square$
Lemma 61.12.10. Let $S$ be a scheme. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site containing $S$. Let $\mathit{Sch}$ be the category of all schemes.
The categories $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$, $S_{pro\text{-}\acute{e}tale}$, and $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ have fibre products agreeing with fibre products in $\mathit{Sch}$.
The categories $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$, $S_{pro\text{-}\acute{e}tale}$ have equalizers agreeing with equalizers in $\mathit{Sch}$.
The categories $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$, and $S_{pro\text{-}\acute{e}tale}$ both have a final object, namely $S/S$.
The category $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ has a final object agreeing with the final object of $\mathit{Sch}$, namely $\mathop{\mathrm{Spec}}(\mathbf{Z})$.
Proof. The category $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ contains $\mathop{\mathrm{Spec}}(\mathbf{Z})$ and is closed under products and fibre products by construction, see Sets, Lemma 3.9.9. Suppose we have $U \to S$, $V \to U$, $W \to U$ morphisms of schemes with $U, V, W \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{pro\text{-}\acute{e}tale})$. The fibre product $V \times _ U W$ in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ is a fibre product in $\mathit{Sch}$ and is the fibre product of $V/S$ with $W/S$ over $U/S$ in the category of all schemes over $S$, and hence also a fibre product in $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$. This proves the result for $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$. If $U \to S$, $V \to U$ and $W \to U$ are weakly étale then so is $V \times _ U W \to S$ (see More on Morphisms, Section 37.64) and hence we get fibre products for $S_{pro\text{-}\acute{e}tale}$. If $U, V, W$ are affine, so is $V \times _ U W$ and hence we get fibre products for $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$.
Let $a, b : U \to V$ be two morphisms in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$. In this case the equalizer of $a$ and $b$ (in the category of schemes) is
\[ V \times _{\Delta _{V/\mathop{\mathrm{Spec}}(\mathbf{Z})}, V \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} V, (a, b)} (U \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} U) \]
which is an object of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ by what we saw above. Thus $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ has equalizers. If $a$ and $b$ are morphisms over $S$, then the equalizer (in the category of schemes) is also given by
\[ V \times _{\Delta _{V/S}, V \times _ S V, (a, b)} (U \times _ S U) \]
hence we see that $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ has equalizers. Moreover, if $U$ and $V$ are weakly-étale over $S$, then so is the equalizer above as a fibre product of schemes weakly étale over $S$. Thus $S_{pro\text{-}\acute{e}tale}$ has equalizers. The statements on final objects is clear. $\square$
Next, we check that the big affine pro-étale site defines the same topos as the big pro-étale site.
Lemma 61.12.11. Let $S$ be a scheme. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site containing $S$. The functor $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is a special cocontinuous functor. Hence it induces an equivalence of topoi from $\mathop{\mathit{Sh}}\nolimits ((\textit{Aff}/S)_{pro\text{-}\acute{e}tale})$ to $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale})$.
Proof. The notion of a special cocontinuous functor is introduced in Sites, Definition 7.29.2. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 7.29.1. Denote the inclusion functor $u : (\textit{Aff}/S)_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$. Being cocontinuous just means that any pro-étale covering of $T/S$, $T$ affine, can be refined by a standard pro-étale covering of $T$. This is the content of Lemma 61.12.5. Hence (1) holds. We see $u$ is continuous simply because a standard pro-étale covering is a pro-étale covering. Hence (2) holds. Parts (3) and (4) follow immediately from the fact that $u$ is fully faithful. And finally condition (5) follows from the fact that every scheme has an affine open covering. $\square$
Lemma 61.12.12. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$. The functor $T_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is cocontinuous and induces a morphism of topoi For a sheaf $\mathcal{G}$ on $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ we have the formula $(i_ f^{-1}\mathcal{G})(U/T) = \mathcal{G}(U/S)$. The functor $i_ f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes with fibre products and equalizers.
\[ i_ f : \mathop{\mathit{Sh}}\nolimits (T_{pro\text{-}\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}) \]
Proof. Denote the functor $u : T_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$. In other words, given a weakly étale morphism $j : U \to T$ corresponding to an object of $T_{pro\text{-}\acute{e}tale}$ we set $u(U \to T) = (f \circ j : U \to S)$. This functor commutes with fibre products, see Lemma 61.12.10. Moreover, $T_{pro\text{-}\acute{e}tale}$ has equalizers and $u$ commutes with them by Lemma 61.12.10. It is clearly cocontinuous. It is also continuous as $u$ transforms coverings to coverings and commutes with fibre products. Hence the lemma follows from Sites, Lemmas 7.21.5 and 7.21.6. $\square$
Lemma 61.12.13. Let $S$ be a scheme. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site containing $S$. The inclusion functor $S_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ satisfies the hypotheses of Sites, Lemma 7.21.8 and hence induces a morphism of sites and a morphism of topoi such that $\pi _ S \circ i_ S = \text{id}$. Moreover, $i_ S = i_{\text{id}_ S}$ with $i_{\text{id}_ S}$ as in Lemma 61.12.12. In particular the functor $i_ S^{-1} = \pi _{S, *}$ is described by the rule $i_ S^{-1}(\mathcal{G})(U/S) = \mathcal{G}(U/S)$.
\[ \pi _ S : (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}\longrightarrow S_{pro\text{-}\acute{e}tale} \]
\[ i_ S : \mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}) \]
Proof. In this case the functor $u : S_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$, in addition to the properties seen in the proof of Lemma 61.12.12 above, also is fully faithful and transforms the final object into the final object. The lemma follows from Sites, Lemma 7.21.8. $\square$
Definition 61.12.14. In the situation of Lemma 61.12.13 the functor $i_ S^{-1} = \pi _{S, *}$ is often called the restriction to the small pro-étale site, and for a sheaf $\mathcal{F}$ on the big pro-étale site we denote $\mathcal{F}|_{S_{pro\text{-}\acute{e}tale}}$ this restriction.
With this notation in place we have for a sheaf $\mathcal{F}$ on the big site and a sheaf $\mathcal{G}$ on the big site that
\begin{align*} \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale})}(\mathcal{F}|_{S_{pro\text{-}\acute{e}tale}}, \mathcal{G}) & = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale})}(\mathcal{F}, i_{S, *}\mathcal{G}) \\ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale})}(\mathcal{G}, \mathcal{F}|_{S_{pro\text{-}\acute{e}tale}}) & = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale})}(\pi _ S^{-1}\mathcal{G}, \mathcal{F}) \end{align*}
Moreover, we have $(i_{S, *}\mathcal{G})|_{S_{pro\text{-}\acute{e}tale}} = \mathcal{G}$ and we have $(\pi _ S^{-1}\mathcal{G})|_{S_{pro\text{-}\acute{e}tale}} = \mathcal{G}$.
Lemma 61.12.15. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$. The functor is cocontinuous, and has a continuous right adjoint They induce the same morphism of topoi We have $f_{big}^{-1}(\mathcal{G})(U/T) = \mathcal{G}(U/S)$. We have $f_{big, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times _ S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.
\[ u : (\mathit{Sch}/T)_{pro\text{-}\acute{e}tale}\longrightarrow (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}, \quad V/T \longmapsto V/S \]
\[ v : (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}\longrightarrow (\mathit{Sch}/T)_{pro\text{-}\acute{e}tale}, \quad (U \to S) \longmapsto (U \times _ S T \to T). \]
\[ f_{big} : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/T)_{pro\text{-}\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}) \]
Proof. The functor $u$ is cocontinuous, continuous, and commutes with fibre products and equalizers (details omitted; compare with proof of Lemma 61.12.12). Hence Sites, Lemmas 7.21.5 and 7.21.6 apply and we deduce the formula for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover, the functor $v$ is a right adjoint because given $U/T$ and $V/S$ we have $\mathop{\mathrm{Mor}}\nolimits _ S(u(U), V) = \mathop{\mathrm{Mor}}\nolimits _ T(U, V \times _ S T)$ as desired. Thus we may apply Sites, Lemmas 7.22.1 and 7.22.2 to get the formula for $f_{big, *}$. $\square$
Lemma 61.12.16. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$.
We have $i_ f = f_{big} \circ i_ T$ with $i_ f$ as in Lemma 61.12.12 and $i_ T$ as in Lemma 61.12.13.
The functor $S_{pro\text{-}\acute{e}tale}\to T_{pro\text{-}\acute{e}tale}$, $(U \to S) \mapsto (U \times _ S T \to T)$ is continuous and induces a morphism of topoi
We have $f_{small, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times _ S T/T)$.
We have a commutative diagram of morphisms of sites
so that $f_{small} \circ \pi _ T = \pi _ S \circ f_{big}$ as morphisms of topoi.
We have $f_{small} = \pi _ S \circ f_{big} \circ i_ T = \pi _ S \circ i_ f$.
\[ f_{small} : \mathop{\mathit{Sh}}\nolimits (T_{pro\text{-}\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale}). \]
\[ \xymatrix{ T_{pro\text{-}\acute{e}tale}\ar[d]_{f_{small}} & (\mathit{Sch}/T)_{pro\text{-}\acute{e}tale}\ar[d]^{f_{big}} \ar[l]^{\pi _ T}\\ S_{pro\text{-}\acute{e}tale}& (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}\ar[l]_{\pi _ S} } \]
Proof. The equality $i_ f = f_{big} \circ i_ T$ follows from the equality $i_ f^{-1} = i_ T^{-1} \circ f_{big}^{-1}$ which is clear from the descriptions of these functors above. Thus we see (1).
The functor $u : S_{pro\text{-}\acute{e}tale}\to T_{pro\text{-}\acute{e}tale}$, $u(U \to S) = (U \times _ S T \to T)$ transforms coverings into coverings and commutes with fibre products, see Lemmas 61.12.4 and 61.12.10. Moreover, both $S_{pro\text{-}\acute{e}tale}$, $T_{pro\text{-}\acute{e}tale}$ have final objects, namely $S/S$ and $T/T$ and $u(S/S) = T/T$. Hence by Sites, Proposition 7.14.7 the functor $u$ corresponds to a morphism of sites $T_{pro\text{-}\acute{e}tale}\to S_{pro\text{-}\acute{e}tale}$. This in turn gives rise to the morphism of topoi, see Sites, Lemma 7.15.2. The description of the pushforward is clear from these references.
Part (3) follows because $\pi _ S$ and $\pi _ T$ are given by the inclusion functors and $f_{small}$ and $f_{big}$ by the base change functors $U \mapsto U \times _ S T$.
Statement (4) follows from (3) by precomposing with $i_ T$. $\square$
In the situation of the lemma, using the terminology of Definition 61.12.14 we have: for $\mathcal{F}$ a sheaf on the big pro-étale site of $T$
61.12.16.1
\begin{equation} \label{proetale-equation-compare-big-small} (f_{big, *}\mathcal{F})|_{S_{pro\text{-}\acute{e}tale}} = f_{small, *}(\mathcal{F}|_{T_{pro\text{-}\acute{e}tale}}), \end{equation}
This equality is clear from the commutativity of the diagram of sites of the lemma, since restriction to the small pro-étale site of $T$, resp. $S$ is given by $\pi _{T, *}$, resp. $\pi _{S, *}$. A similar formula involving pullbacks and restrictions is false.
Lemma 61.12.17. Given schemes $X$, $Y$, $Y$ in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ and morphisms $f : X \to Y$, $g : Y \to Z$ we have $g_{big} \circ f_{big} = (g \circ f)_{big}$ and $g_{small} \circ f_{small} = (g \circ f)_{small}$.
Proof. This follows from the simple description of pushforward and pullback for the functors on the big sites from Lemma 61.12.15. For the functors on the small sites this follows from the description of the pushforward functors in Lemma 61.12.16. $\square$
Lemma 61.12.18. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site. Consider a cartesian diagram in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$. Then $i_ g^{-1} \circ f_{big, *} = f'_{small, *} \circ (i_{g'})^{-1}$ and $g_{big}^{-1} \circ f_{big, *} = f'_{big, *} \circ (g'_{big})^{-1}$.
\[ \xymatrix{ T' \ar[r]_{g'} \ar[d]_{f'} & T \ar[d]^ f \\ S' \ar[r]^ g & S } \]
Proof. Since the diagram is cartesian, we have for $U'/S'$ that $U' \times _{S'} T' = U' \times _ S T$. Hence both $i_ g^{-1} \circ f_{big, *}$ and $f'_{small, *} \circ (i_{g'})^{-1}$ send a sheaf $\mathcal{F}$ on $(\mathit{Sch}/T)_{pro\text{-}\acute{e}tale}$ to the sheaf $U' \mapsto \mathcal{F}(U' \times _{S'} T')$ on $S'_{pro\text{-}\acute{e}tale}$ (use Lemmas 61.12.12 and 61.12.15). The second equality can be proved in the same manner or can be deduced from the very general Sites, Lemma 7.28.1. $\square$
We can think about a sheaf on the big pro-étale site of $S$ as a collection of sheaves on the small pro-étale site on schemes over $S$.
Lemma 61.12.19. Let $S$ be a scheme contained in a big pro-étale site $\mathit{Sch}_{pro\text{-}\acute{e}tale}$. A sheaf $\mathcal{F}$ on the big pro-étale site $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is given by the following data:
for every $T/S \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale})$ a sheaf $\mathcal{F}_ T$ on $T_{pro\text{-}\acute{e}tale}$,
for every $f : T' \to T$ in $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ a map $c_ f : f_{small}^{-1}\mathcal{F}_ T \to \mathcal{F}_{T'}$.
These data are subject to the following conditions:
given any $f : T' \to T$ and $g : T'' \to T'$ in $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ the composition $c_ g \circ g_{small}^{-1}c_ f$ is equal to $c_{f \circ g}$, and
if $f : T' \to T$ in $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is weakly étale then $c_ f$ is an isomorphism.
Proof. Identical to the proof of Topologies, Lemma 34.4.20. $\square$
Lemma 61.12.20. Let $S$ be a scheme. Let $S_{affine, {pro\text{-}\acute{e}tale}}$ denote the full subcategory of $S_{pro\text{-}\acute{e}tale}$ consisting of affine objects. A covering of $S_{affine, {pro\text{-}\acute{e}tale}}$ will be a standard pro-étale covering, see Definition 61.12.6. Then restriction defines an equivalence of topoi $\mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale}) \cong \mathop{\mathit{Sh}}\nolimits (S_{affine, {pro\text{-}\acute{e}tale}})$.
\[ \mathcal{F} \longmapsto \mathcal{F}|_{S_{affine, {\acute{e}tale}}} \]
Proof. This you can show directly from the definitions, and is a good exercise. But it also follows immediately from Sites, Lemma 7.29.1 by checking that the inclusion functor $S_{affine, {pro\text{-}\acute{e}tale}} \to S_{pro\text{-}\acute{e}tale}$ is a special cocontinuous functor (see Sites, Definition 7.29.2). $\square$
Lemma 61.12.21. Let $S$ be an affine scheme. Let $S_{app}$ denote the full subcategory of $S_{pro\text{-}\acute{e}tale}$ consisting of affine objects $U$ such that $\mathcal{O}(S) \to \mathcal{O}(U)$ is ind-étale. A covering of $S_{app}$ will be a standard pro-étale covering, see Definition 61.12.6. Then restriction defines an equivalence of topoi $\mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale}) \cong \mathop{\mathit{Sh}}\nolimits (S_{app})$.
\[ \mathcal{F} \longmapsto \mathcal{F}|_{S_{app}} \]
Proof. By Lemma 61.12.20 we may replace $S_{pro\text{-}\acute{e}tale}$ by $S_{affine, {pro\text{-}\acute{e}tale}}$. The lemma follows from Sites, Lemma 7.29.1 by checking that the inclusion functor $S_{app} \to S_{affine, {pro\text{-}\acute{e}tale}}$ is a special cocontinuous functor, see Sites, Definition 7.29.2. The conditions of Sites, Lemma 7.29.1 follow immediately from the definition and the facts (a) any object $U$ of $S_{affine, {pro\text{-}\acute{e}tale}}$ has a covering $\{ V \to U\} $ with $V$ ind-étale over $X$ (Proposition 61.9.1) and (b) the functor $u$ is fully faithful. $\square$
Lemma 61.12.22. Let $S$ be a scheme. The topology on each of the pro-étale sites $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, $S_{pro\text{-}\acute{e}tale}$, $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$, $S_{affine, {pro\text{-}\acute{e}tale}}$, and $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ is subcanonical.
Proof. Combine Lemma 61.12.2 and Descent, Lemma 35.13.7. $\square$
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