## Tag `0988`

## 52.11. The pro-étale site

The (small) pro-étale site of a scheme has some remarkable properties. In particular, it has enough w-contractible objects which implies a number of useful consequences for the derived category of abelian sheaves and for inverse systems of sheaves. Thus it is well adapted to deal with some of the intricacies of working with $\ell$-adic sheaves.

On the other hand, the pro-étale topology is a bit like the fpqc topology (see Topologies, Section 33.9) 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

thepro-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.Another curiosity is that we define pro-étale coverings using weakly étale morphisms of schemes, see More on Morphisms, Section 36.53. 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 52.9.1 assures us that we obtain the same sheaves with the definition that follows.

Definition 52.11.1. 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 $\bigcup_{j = 1}^n f_{a(j)}(V_j) = U$.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 52.11.2. 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

- (5) 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 33.9.2. $\square$Lemma 52.11.3. Any étale covering and any Zariski covering is a pro-étale covering.

Proof.This follows from the corresponding result for fpqc coverings (Topologies, Lemma 33.9.6), Lemma 52.11.2, and the fact that an étale morphism is a weakly étale morphism, see More on Morphisms, Lemma 36.53.9. $\square$Lemma 52.11.4. 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 36.53.5 and 36.53.6), Lemma 52.11.2, and the corresponding results for fpqc coverings, see Topologies, Lemma 33.9.7. $\square$Lemma 52.11.5. 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 52.11.6. Let $T$ be an affine scheme. A

standard pro-étale coveringof $T$ 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 interrupt the discussion for an explanation of the notion of w-contractible rings in terms of pro-étale coverings.

Lemma 52.11.7. Let $T = \mathop{\rm Spec}(A)$ be an affine scheme. The following are equivalent

- $A$ is w-contractible, and
- every pro-étale covering of $T$ can be refined by a Zariski covering of the form $T = \coprod_{i = 1, \ldots, n} U_i$.

Proof.Assume $A$ is w-contractible. By Lemma 52.11.5 it suffices to prove we can refine every standard pro-étale covering $\{f_i : T_i \to T\}_{i = 1, \ldots, n}$ by a Zariski covering of $T$. The morphism $\coprod T_i \to T$ is a surjective weakly étale morphism of affine schemes. Hence by Definition 52.10.1 there exists a morphism $\sigma : T \to \coprod T_i$ over $T$. Then the Zariski covering $T = \coprod \sigma^{-1}(T_i)$ refines $\{f_i : T_i \to T\}$.Conversely, assume (2). If $A \to B$ is faithfully flat and weakly étale, then $\{\mathop{\rm Spec}(B) \to T\}$ is a pro-étale covering. Hence there exists a Zariski covering $T = \coprod U_i$ and morphisms $U_i \to \mathop{\rm Spec}(B)$ over $T$. Since $T = \coprod U_i$ we obtain $T \to \mathop{\rm Spec}(B)$, i.e., an $A$-algebra map $B \to A$. This means $A$ is w-contractible. $\square$

We follow the general outline given in Topologies, Section 33.2 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 52.11.8. A

big pro-étale siteis any site $\textit{Sch}_{pro\text{-}\acute{e}tale}$ 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 $$ Bound(\kappa) = \max\{\kappa^{2^{2^{2^\kappa}}}, \kappa^{\aleph_0}, \kappa^+\}. $$
- As underlying category take any category $\textit{Sch}_\alpha$ 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 $\textit{Sch}_\alpha$ and the class of pro-étale coverings, and the set $\text{Cov}_0$ chosen above.

See the remarks following Topologies, Definition 33.3.5 for motivation and explanation regarding the definition of big sites.

Before we continue with the introduction of the big and small pro-étale sites of a scheme, let us point out that (1) our category contains many weakly contractible objects, and (2) the topology on a big pro-étale site $\textit{Sch}_{pro\text{-}\acute{e}tale}$ is in some sense induced from the pro-étale topology on the category of all schemes.

Lemma 52.11.9. Let $\textit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site as in Definition 52.11.8. Let $T = \mathop{\rm Spec}(A)$ be an affine object of $\textit{Sch}_{pro\text{-}\acute{e}tale}$. If $A$ is w-contractible, then $T$ is a weakly contractible (Sites, Definition 7.39.2) object of $\textit{Sch}_{pro\text{-}\acute{e}tale}$.

Proof.Let $\mathcal{F} \to \mathcal{G}$ be a surjection of sheaves on $\textit{Sch}_{pro\text{-}\acute{e}tale}$. Let $s \in \mathcal{G}(T)$. We have to show that $s$ is in the image of $\mathcal{F}(T) \to \mathcal{G}(T)$. We can find a covering $\{T_i \to T\}$ of $\textit{Sch}_{pro\text{-}\acute{e}tale}$ such that $s$ lifts to a section of $\mathcal{F}$ over $T_i$ (Sites, Definition 7.11.1). By Lemma 52.11.7 we can refine $\{T_i \to T\}$ by a Zariski covering of the form $T = \coprod_{j = 1, \ldots, m} V_j$. Hence we get $t_j \in \mathcal{F}(U_j)$ mapping to $s|_{U_j}$. Since Zariski coverings are coverings in $\textit{Sch}_{pro\text{-}\acute{e}tale}$ (Lemma 52.11.3) we conclude that $\mathcal{F}(T) = \prod \mathcal{F}(U_j)$. Thus, taking $t = (t_1, \ldots, t_m) \in \mathcal{F}(T)$ is a section mapping to $s$. $\square$Lemma 52.11.10. Let $\textit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site as in Definition 52.11.8. For every object $T$ of $\textit{Sch}_{pro\text{-}\acute{e}tale}$ there exists a covering $\{T_i \to T\}$ in $\textit{Sch}_{pro\text{-}\acute{e}tale}$ with each $T_i$ affine and the spectrum of a w-contractible ring. In particular, $T_i$ is weakly contractible in $\textit{Sch}_{pro\text{-}\acute{e}tale}$.

Proof.For those readers who do not care about set-theoretical issues this lemma is a trivial consequence of Lemma 52.11.9 and Proposition 52.10.3. Here are the details. Choose an affine open covering $T = \bigcup U_i$. Write $U_i = \mathop{\rm Spec}(A_i)$. Choose faithfully flat, ind-étale ring maps $A_i \to D_i$ such that $D_i$ is w-contractible as in Proposition 52.10.3. The family of morphisms $\{\mathop{\rm Spec}(D_i) \to T\}$ is a pro-étale covering. If we can show that $\mathop{\rm Spec}(D_i)$ is isomorphic to an object, say $T_i$, of $\textit{Sch}_{pro\text{-}\acute{e}tale}$, then $\{T_i \to T\}$ will be combinatorially equivalent to a covering of $\textit{Sch}_{pro\text{-}\acute{e}tale}$ by the construction of $\textit{Sch}_{pro\text{-}\acute{e}tale}$ in Definition 52.11.8 and more precisely the application of Sets, Lemma 3.11.1 in the last step. To prove $\mathop{\rm Spec}(D_i)$ is isomorphic to an object of $\textit{Sch}_{pro\text{-}\acute{e}tale}$, it suffices to prove that $|D_i| \leq Bound(\text{Size}(T))$ by the construction of $\textit{Sch}_{pro\text{-}\acute{e}tale}$ in Definition 52.11.8 and more precisely the application of Sets, Lemma 3.9.2 in step (3). Since $|A_i| \leq \text{size}(U_i) \leq \text{size}(T)$ by Sets, Lemmas 3.9.4 and 3.9.7 we get $|D_i| \leq \kappa^{2^{2^{2^\kappa}}}$ where $\kappa = \text{size}(T)$ by Remark 52.10.4. Thus by our choice of the function $Bound$ in Definition 52.11.8 we win. $\square$Lemma 52.11.11. Let $\textit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site as in Definition 52.11.8. Let $T \in \mathop{\rm Ob}\nolimits(\textit{Sch}_{pro\text{-}\acute{e}tale})$. Let $\{T_i \to T\}_{i \in I}$ be an arbitrary pro-étale covering of $T$. There exists a covering $\{U_j \to T\}_{j \in J}$ of $T$ in the site $\textit{Sch}_{pro\text{-}\acute{e}tale}$ which refines $\{T_i \to T\}_{i \in I}$.

Proof.Namely, we first let $\{V_k \to T\}$ be a covering as in Lemma 52.11.10. Then the pro-étale coverings $\{T_i \times_T V_k \to V_k\}$ can be refined by a finite disjoint open covering $V_k = V_{k, 1} \amalg \ldots \amalg V_{k, n_k}$, see Lemma 52.11.7. Then $\{V_{k, i} \to T\}$ is a covering of $\textit{Sch}_{pro\text{-}\acute{e}tale}$ which refines $\{T_i \to T\}_{i \in I}$. $\square$Definition 52.11.12. Let $S$ be a scheme. Let $\textit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site containing $S$.

- The
big pro-étale site of $S$, denoted $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$, is the site $\textit{Sch}_{pro\text{-}\acute{e}tale}/S$ introduced in Sites, Section 7.24.- The
small pro-étale site of $S$, which we denote $S_{pro\text{-}\acute{e}tale}$, is the full subcategory of $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$ 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 $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$ with $U \in \mathop{\rm Ob}\nolimits(S_{pro\text{-}\acute{e}tale})$.- The
big affine pro-étale site of $S$, denoted $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$, is the full subcategory of $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$ 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 $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$ 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 52.11.13. Let $S$ be a scheme. Let $\textit{Sch}_{pro\text{-}\acute{e}tale}$ 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 $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is a site, it suffices to prove that given any covering $\{U_i \to U\}$ of $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$ with $U \in \mathop{\rm Ob}\nolimits(S_{pro\text{-}\acute{e}tale})$ we also have $U_i \in \mathop{\rm Ob}\nolimits(S_{pro\text{-}\acute{e}tale})$. 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 52.11.2 and the corresponding result for standard fpqc coverings (Topologies, Lemma 33.9.10). $\square$

Lemma 52.11.14. Let $S$ be a scheme. Let $\textit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site containing $S$. Let $\textit{Sch}$ be the category of all schemes.

- The categories $\textit{Sch}_{pro\text{-}\acute{e}tale}$, $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$, $S_{pro\text{-}\acute{e}tale}$, and $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ have fibre products agreeing with fibre products in $\textit{Sch}$.
- The categories $\textit{Sch}_{pro\text{-}\acute{e}tale}$, $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$, $S_{pro\text{-}\acute{e}tale}$ have equalizers agreeing with equalizers in $\textit{Sch}$.
- The categories $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$, and $S_{pro\text{-}\acute{e}tale}$ both have a final object, namely $S/S$.
- The category $\textit{Sch}_{pro\text{-}\acute{e}tale}$ has a final object agreeing with the final object of $\textit{Sch}$, namely $\mathop{\rm Spec}(\mathbf{Z})$.

Proof.The category $\textit{Sch}_{pro\text{-}\acute{e}tale}$ contains $\mathop{\rm Spec}(\mathbf{Z})$ 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 $U, V, W \in \mathop{\rm Ob}\nolimits(\textit{Sch}_{pro\text{-}\acute{e}tale})$. The fibre product $V \times_U W$ in $\textit{Sch}_{pro\text{-}\acute{e}tale}$ is a fibre product in $\textit{Sch}$ 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 $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$. This proves the result for $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$. 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 36.53) 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 $\textit{Sch}_{pro\text{-}\acute{e}tale}$. In this case the equalizer of $a$ and $b$ (in the category of schemes) is $$ V \times_{\Delta_{V/\mathop{\rm Spec}(\mathbf{Z})}, V \times_{\mathop{\rm Spec}(\mathbf{Z})} V, (a, b)} (U \times_{\mathop{\rm Spec}(\mathbf{Z})} U) $$ which is an object of $\textit{Sch}_{pro\text{-}\acute{e}tale}$ by what we saw above. Thus $\textit{Sch}_{pro\text{-}\acute{e}tale}$ 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 $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$ 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 52.11.15. Let $S$ be a scheme. Let $\textit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site containing $S$. The functor $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale} \to (\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is a special cocontinuous functor. Hence it induces an equivalence of topoi from $\mathop{\textit{Sh}}\nolimits((\textit{Aff}/S)_{pro\text{-}\acute{e}tale})$ to $\mathop{\textit{Sh}}\nolimits((\textit{Sch}/S)_{pro\text{-}\acute{e}tale})$.

Proof.The notion of a special cocontinuous functor is introduced in Sites, Definition 7.28.2. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 7.28.1. Denote the inclusion functor $u : (\textit{Aff}/S)_{pro\text{-}\acute{e}tale} \to (\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$. 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 52.11.5. 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 52.11.16. Let $\textit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site. Let $f : T \to S$ be a morphism in $\textit{Sch}_{pro\text{-}\acute{e}tale}$. The functor $T_{pro\text{-}\acute{e}tale} \to (\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is cocontinuous and induces a morphism of topoi $$ i_f : \mathop{\textit{Sh}}\nolimits(T_{pro\text{-}\acute{e}tale}) \longrightarrow \mathop{\textit{Sh}}\nolimits((\textit{Sch}/S)_{pro\text{-}\acute{e}tale}) $$ For a sheaf $\mathcal{G}$ on $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$ 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.

Proof.Denote the functor $u : T_{pro\text{-}\acute{e}tale} \to (\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$. 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 52.11.14. Moreover, $T_{pro\text{-}\acute{e}tale}$ has equalizers and $u$ commutes with them by Lemma 52.11.14. 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 7.20.5 and 7.20.6. $\square$Lemma 52.11.17. Let $S$ be a scheme. Let $\textit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site containing $S$. The inclusion functor $S_{pro\text{-}\acute{e}tale} \to (\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$ satisfies the hypotheses of Sites, Lemma 7.20.8 and hence induces a morphism of sites $$ \pi_S : (\textit{Sch}/S)_{pro\text{-}\acute{e}tale} \longrightarrow S_{pro\text{-}\acute{e}tale} $$ and a morphism of topoi $$ i_S : \mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale}) \longrightarrow \mathop{\textit{Sh}}\nolimits((\textit{Sch}/S)_{pro\text{-}\acute{e}tale}) $$ such that $\pi_S \circ i_S = \text{id}$. Moreover, $i_S = i_{\text{id}_S}$ with $i_{\text{id}_S}$ as in Lemma 52.11.16. In particular the functor $i_S^{-1} = \pi_{S, *}$ is described by the rule $i_S^{-1}(\mathcal{G})(U/S) = \mathcal{G}(U/S)$.

Proof.In this case the functor $u : S_{pro\text{-}\acute{e}tale} \to (\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$, in addition to the properties seen in the proof of Lemma 52.11.16 above, also is fully faithful and transforms the final object into the final object. The lemma follows from Sites, Lemma 7.20.8. $\square$Definition 52.11.18. In the situation of Lemma 52.11.17 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{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale})}(\mathcal{F}|_{S_{pro\text{-}\acute{e}tale}}, \mathcal{G}) & = \mathop{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits((\textit{Sch}/S)_{pro\text{-}\acute{e}tale})}(\mathcal{F}, i_{S, *}\mathcal{G}) \\ \mathop{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale})}(\mathcal{G}, \mathcal{F}|_{S_{pro\text{-}\acute{e}tale}}) & = \mathop{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits((\textit{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 52.11.19. Let $\textit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site. Let $f : T \to S$ be a morphism in $\textit{Sch}_{pro\text{-}\acute{e}tale}$. The functor $$ u : (\textit{Sch}/T)_{pro\text{-}\acute{e}tale} \longrightarrow (\textit{Sch}/S)_{pro\text{-}\acute{e}tale}, \quad V/T \longmapsto V/S $$ is cocontinuous, and has a continuous right adjoint $$ v : (\textit{Sch}/S)_{pro\text{-}\acute{e}tale} \longrightarrow (\textit{Sch}/T)_{pro\text{-}\acute{e}tale}, \quad (U \to S) \longmapsto (U \times_S T \to T). $$ They induce the same morphism of topoi $$ f_{big} : \mathop{\textit{Sh}}\nolimits((\textit{Sch}/T)_{pro\text{-}\acute{e}tale}) \longrightarrow \mathop{\textit{Sh}}\nolimits((\textit{Sch}/S)_{pro\text{-}\acute{e}tale}) $$ 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.

Proof.The functor $u$ is cocontinuous, continuous, and commutes with fibre products and equalizers (details omitted; compare with proof of Lemma 52.11.16). Hence Sites, Lemmas 7.20.5 and 7.20.6 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 $\mathop{\rm Mor}\nolimits_S(u(U), V) = \mathop{\rm Mor}\nolimits_T(U, V \times_S T)$ as desired. Thus we may apply Sites, Lemmas 7.21.1 and 7.21.2 to get the formula for $f_{big, *}$. $\square$Lemma 52.11.20. Let $\textit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site. Let $f : T \to S$ be a morphism in $\textit{Sch}_{pro\text{-}\acute{e}tale}$.

- We have $i_f = f_{big} \circ i_T$ with $i_f$ as in Lemma 52.11.16 and $i_T$ as in Lemma 52.11.17.
- 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 $$ f_{small} : \mathop{\textit{Sh}}\nolimits(T_{pro\text{-}\acute{e}tale}) \longrightarrow \mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale}). $$ We have $f_{small, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times_S T/T)$.
- We have a commutative diagram of morphisms of sites $$ \xymatrix{ T_{pro\text{-}\acute{e}tale} \ar[d]_{f_{small}} & (\textit{Sch}/T)_{pro\text{-}\acute{e}tale} \ar[d]^{f_{big}} \ar[l]^{\pi_T}\\ S_{pro\text{-}\acute{e}tale} & (\textit{Sch}/S)_{pro\text{-}\acute{e}tale} \ar[l]_{\pi_S} } $$ so that $f_{small} \circ \pi_T = \pi_S \circ f_{big}$ as morphisms of topoi.
- We have $f_{small} = \pi_S \circ f_{big} \circ i_T = \pi_S \circ i_f$.

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 52.11.4 and 52.11.14. 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 7.14.6 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 52.11.18 we have: for $\mathcal{F}$ a sheaf on the big pro-étale site of $T$ $$ (f_{big, *}\mathcal{F})|_{S_{pro\text{-}\acute{e}tale}} = f_{small, *}(\mathcal{F}|_{T_{pro\text{-}\acute{e}tale}}), $$ 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 52.11.21. Given schemes $X$, $Y$, $Y$ in $\textit{Sch}_{pro\text{-}\acute{e}tale}$ 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 52.11.19. For the functors on the small sites this follows from the description of the pushforward functors in Lemma 52.11.20. $\square$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 52.11.22. Let $S$ be a scheme contained in a big pro-étale site $\textit{Sch}_{pro\text{-}\acute{e}tale}$. A sheaf $\mathcal{F}$ on the big pro-étale site $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is given by the following data:

- for every $T/S \in \mathop{\rm Ob}\nolimits((\textit{Sch}/S)_{pro\text{-}\acute{e}tale})$ a sheaf $\mathcal{F}_T$ on $T_{pro\text{-}\acute{e}tale}$,
- for every $f : T' \to T$ in $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$ a map $c_f : f_{small}^{-1}\mathcal{F}_T \to \mathcal{F}_{T'}$.
These data are subject to the following conditions:

- (a) given any $f : T' \to T$ and $g : T'' \to T'$ in $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$ the composition $g_{small}^{-1}c_f \circ c_g$ is equal to $c_{f \circ g}$, and
- (b) if $f : T' \to T$ in $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is weakly étale then $c_f$ is an isomorphism.

Proof.Identical to the proof of Topologies, Lemma 33.4.19. $\square$Lemma 52.11.23. 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 étale covering, see Definition 52.11.6. Then restriction $$ \mathcal{F} \longmapsto \mathcal{F}|_{S_{affine, {\acute{e}tale}}} $$ defines an equivalence of topoi $\mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale}) \cong \mathop{\textit{Sh}}\nolimits(S_{affine, {pro\text{-}\acute{e}tale}})$.

Proof.This you can show directly from the definitions, and is a good exercise. But it also follows immediately from Sites, Lemma 7.28.1 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 7.28.2). $\square$Lemma 52.11.24. 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 52.11.6. Then restriction $$ \mathcal{F} \longmapsto \mathcal{F}|_{S_{app}} $$ defines an equivalence of topoi $\mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale}) \cong \mathop{\textit{Sh}}\nolimits(S_{app})$.

Proof.By Lemma 52.11.23 we may replace $S_{pro\text{-}\acute{e}tale}$ by $S_{affine, {pro\text{-}\acute{e}tale}}$. The lemma follows from Sites, Lemma 7.28.1 by checking that the inclusion functor $S_{app} \to S_{affine, {pro\text{-}\acute{e}tale}}$ is a special cocontinuous functor, see Sites, Definition 7.28.2. The conditions of Sites, Lemma 7.28.1 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 52.9.1) and (b) the functor $u$ is fully faithful. $\square$Next we show that cohomology of sheaves is independent of the choice of a partial universe. Namely, the functor $g_*$ of the lemma below is an embedding of pro-étale topoi which does not change cohomology.

Lemma 52.11.25. Let $S$ be a scheme. Let $S_{pro\text{-}\acute{e}tale} \subset S_{pro\text{-}\acute{e}tale}'$ be two small pro-étale sites of $S$ as constructed in Definition 52.11.12. Then the inclusion functor satisfies the assumptions of Sites, Lemma 7.20.8. Hence there exist morphisms of topoi $$ \xymatrix{ \mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale}) \ar[r]^g & \mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale}') \ar[r]^f & \mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale}) } $$ whose composition is isomorphic to the identity and with $f_* = g^{-1}$. Moreover,

- for $\mathcal{F}' \in \textit{Ab}(S_{pro\text{-}\acute{e}tale}')$ we have $H^p(S_{pro\text{-}\acute{e}tale}', \mathcal{F}') = H^p(S_{pro\text{-}\acute{e}tale}, g^{-1}\mathcal{F}')$,
- for $\mathcal{F} \in \textit{Ab}(S_{pro\text{-}\acute{e}tale})$ we have $$ H^p(S_{pro\text{-}\acute{e}tale}, \mathcal{F}) = H^p(S_{pro\text{-}\acute{e}tale}', g_*\mathcal{F}) = H^p(S_{pro\text{-}\acute{e}tale}', f^{-1}\mathcal{F}). $$

Proof.The inclusion functor is fully faithful and continuous. We have seen that $S_{pro\text{-}\acute{e}tale}$ and $S_{pro\text{-}\acute{e}tale}'$ have fibre products and final objects and that our functor commutes with these (Lemma 52.11.14). It follows from Lemma 52.11.11 that the inclusion functor is cocontinuous. Hence the existence of $f$ and $g$ follows from Sites, Lemma 7.20.8. The equality in (1) is Cohomology on Sites, Lemma 21.8.2. Part (2) follows from (1) as $\mathcal{F} = g^{-1}g_*\mathcal{F} = g^{-1}f^{-1}\mathcal{F}$. $\square$Lemma 52.11.26. Let $S$ be a scheme. The topology on each of the pro-étale sites $S_{pro\text{-}\acute{e}tale}$, $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$, $S_{affine, {pro\text{-}\acute{e}tale}}$, and $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ is subcanonical.

Proof.Combine Lemma 52.11.2 and Descent, Lemma 34.10.3. $\square$Lemma 52.11.27. Let $S$ be a scheme. The pro-étale sites $S_{pro\text{-}\acute{e}tale}$, $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$, $S_{affine, {pro\text{-}\acute{e}tale}}$, and $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ and if $S$ is affine $S_{app}$ have enough quasi-compact, weakly contractible objects, see Sites, Definition 7.39.2.

Proof.Follows immediately from Lemma 52.11.10. $\square$

The code snippet corresponding to this tag is a part of the file `proetale.tex` and is located in lines 1840–2672 (see updates for more information).

```
\section{The pro-\'etale site}
\label{section-proetale}
\noindent
The (small) pro-\'etale site of a scheme has some remarkable properties.
In particular, it has enough w-contractible objects which implies
a number of useful consequences for the derived category
of abelian sheaves and for inverse systems of sheaves. Thus it is
well adapted to deal with some of the intricacies of working
with $\ell$-adic sheaves.
\medskip\noindent
On the other hand, the pro-\'etale topology is a bit like
the fpqc topology (see Topologies, Section \ref{topologies-section-fpqc})
in that the topos of sheaves on the small pro-\'etale site of a scheme
depends on the choice of the underlying category of schemes. Thus we cannot
speak of {\it the} pro-\'etale 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.
\medskip\noindent
Another curiosity is that we define pro-\'etale coverings using weakly
\'etale morphisms of schemes, see
More on Morphisms, Section \ref{more-morphisms-section-weakly-etale}.
The reason is that, on the one hand, it is somewhat awkward to define
the notion of a pro-\'etale morphism of schemes, and on the other,
Proposition \ref{proposition-weakly-etale}
assures us that we obtain the same sheaves with the
definition that follows.
\begin{definition}
\label{definition-fpqc-covering}
Let $T$ be a scheme. A {\it pro-\'etale 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-\'etale 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 $\bigcup_{j = 1}^n f_{a(j)}(V_j) = U$.
\end{definition}
\noindent
To be sure this condition implies that $T = \bigcup f_i(T_i)$.
Here is a lemma that will allow us to recognize pro-\'etale coverings.
It will also allow us to reduce many lemmas about pro-\'etale coverings
to the corresponding results for fpqc coverings.
\begin{lemma}
\label{lemma-recognize-proetale-covering}
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
\begin{enumerate}
\item $\{f_i : T_i \to T\}_{i \in I}$ is a pro-\'etale covering,
\item each $f_i$ is weakly \'etale and $\{f_i : T_i \to T\}_{i \in I}$
is an fpqc covering,
\item each $f_i$ is weakly \'etale 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 $U = \bigcup f_i(U_i)$,
\item each $f_i$ is weakly \'etale 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
$U_\alpha =
\bigcup_{j = 1, \ldots, n(\alpha)} f_{i_{\alpha, j}}(U_{\alpha, j})$.
\end{enumerate}
If $T$ is quasi-separated, these are also equivalent to
\begin{enumerate}
\item[(5)] each $f_i$ is weakly \'etale, 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 $T$.
\end{enumerate}
\end{lemma}
\begin{proof}
The equivalence of (1) and (2) is immediate from the definitions.
Hence the lemma follows from
Topologies, Lemma \ref{topologies-lemma-recognize-fpqc-covering}.
\end{proof}
\begin{lemma}
\label{lemma-etale-proetale}
Any \'etale covering and any Zariski covering is a pro-\'etale covering.
\end{lemma}
\begin{proof}
This follows from the corresponding result for fpqc coverings
(Topologies, Lemma
\ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc}),
Lemma \ref{lemma-recognize-proetale-covering}, and
the fact that an \'etale morphism is a weakly \'etale morphism, see
More on Morphisms, Lemma \ref{more-morphisms-lemma-when-weakly-etale}.
\end{proof}
\begin{lemma}
\label{lemma-proetale}
Let $T$ be a scheme.
\begin{enumerate}
\item If $T' \to T$ is an isomorphism then $\{T' \to T\}$
is a pro-\'etale covering of $T$.
\item If $\{T_i \to T\}_{i\in I}$ is a pro-\'etale covering and for each
$i$ we have a pro-\'etale 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-\'etale covering.
\item If $\{T_i \to T\}_{i\in I}$ is a pro-\'etale covering
and $T' \to T$ is a morphism of schemes then
$\{T' \times_T T_i \to T'\}_{i\in I}$ is a pro-\'etale covering.
\end{enumerate}
\end{lemma}
\begin{proof}
This follows from the fact that composition and base changes
of weakly \'etale morphisms are weakly \'etale
(More on Morphisms, Lemmas
\ref{more-morphisms-lemma-composition-weakly-etale} and
\ref{more-morphisms-lemma-base-change-weakly-etale}),
Lemma \ref{lemma-recognize-proetale-covering}, and
the corresponding results for fpqc coverings, see
Topologies, Lemma \ref{topologies-lemma-fpqc}.
\end{proof}
\begin{lemma}
\label{lemma-proetale-affine}
Let $T$ be an affine scheme. Let $\{T_i \to T\}_{i \in I}$ be a pro-\'etale
covering of $T$. Then there exists a pro-\'etale 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 $T_i$.
\end{lemma}
\begin{proof}
This follows directly from the definition.
\end{proof}
\noindent
Thus we define the corresponding standard coverings of affines as follows.
\begin{definition}
\label{definition-standard-proetale}
Let $T$ be an affine scheme. A {\it standard pro-\'etale 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 \'etale, and
$T = \bigcup f_i(T_i)$.
\end{definition}
\noindent
We interrupt the discussion for an explanation of the notion
of w-contractible rings in terms of pro-\'etale coverings.
\begin{lemma}
\label{lemma-w-contractible-proetale-cover}
Let $T = \Spec(A)$ be an affine scheme. The following are equivalent
\begin{enumerate}
\item $A$ is w-contractible, and
\item every pro-\'etale covering of $T$ can be refined by
a Zariski covering of the form $T = \coprod_{i = 1, \ldots, n} U_i$.
\end{enumerate}
\end{lemma}
\begin{proof}
Assume $A$ is w-contractible. By Lemma \ref{lemma-proetale-affine}
it suffices to prove we can refine every standard pro-\'etale covering
$\{f_i : T_i \to T\}_{i = 1, \ldots, n}$ by a Zariski covering of $T$.
The morphism $\coprod T_i \to T$ is a surjective weakly \'etale morphism
of affine schemes. Hence by Definition \ref{definition-w-contractible}
there exists a morphism $\sigma : T \to \coprod T_i$ over $T$.
Then the Zariski covering $T = \coprod \sigma^{-1}(T_i)$
refines $\{f_i : T_i \to T\}$.
\medskip\noindent
Conversely, assume (2). If $A \to B$ is faithfully flat and weakly \'etale,
then $\{\Spec(B) \to T\}$ is a pro-\'etale covering.
Hence there exists a Zariski covering $T = \coprod U_i$
and morphisms $U_i \to \Spec(B)$ over $T$. Since $T = \coprod U_i$
we obtain $T \to \Spec(B)$, i.e., an $A$-algebra map $B \to A$.
This means $A$ is w-contractible.
\end{proof}
\noindent
We follow the general outline given in
Topologies, Section \ref{topologies-section-procedure}
for constructing the big pro-\'etale 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.
\begin{definition}
\label{definition-big-proetale-site}
A {\it big pro-\'etale site} is any site $\Sch_\proetale$ as in
Sites, Definition \ref{sites-definition-site} constructed as follows:
\begin{enumerate}
\item Choose any set of schemes $S_0$, and any set of pro-\'etale coverings
$\text{Cov}_0$ among these schemes.
\item Change the function $Bound$ of
Sets, Equation (\ref{sets-equation-bound}) into
$$
Bound(\kappa) = \max\{\kappa^{2^{2^{2^\kappa}}}, \kappa^{\aleph_0}, \kappa^+\}.
$$
\item As underlying category take any category $\Sch_\alpha$
constructed as in Sets, Lemma \ref{sets-lemma-construct-category}
starting with the set $S_0$ and the function $Bound$.
\item Choose any set of coverings as in
Sets, Lemma \ref{sets-lemma-coverings-site} starting with the
category $\Sch_\alpha$ and the class of pro-\'etale coverings,
and the set $\text{Cov}_0$ chosen above.
\end{enumerate}
\end{definition}
\noindent
See the remarks following
Topologies, Definition \ref{topologies-definition-big-zariski-site}
for motivation and explanation regarding the definition of big sites.
\medskip\noindent
Before we continue with the introduction of the big and small
pro-\'etale sites of a scheme, let us point out that (1) our category
contains many weakly contractible objects, and (2) the topology on a
big pro-\'etale site $\Sch_\proetale$ is in some sense induced from
the pro-\'etale topology on the category of all schemes.
\begin{lemma}
\label{lemma-w-contractible-is-weakly-contractible}
Let $\Sch_\proetale$ be a big pro-\'etale site as in
Definition \ref{definition-big-proetale-site}.
Let $T = \Spec(A)$ be an affine object of $\Sch_\proetale$.
If $A$ is w-contractible, then $T$ is a weakly contractible
(Sites, Definition \ref{sites-definition-w-contractible})
object of $\Sch_\proetale$.
\end{lemma}
\begin{proof}
Let $\mathcal{F} \to \mathcal{G}$ be a surjection of sheaves on
$\Sch_\proetale$. Let $s \in \mathcal{G}(T)$. We have to show that
$s$ is in the image of $\mathcal{F}(T) \to \mathcal{G}(T)$. We can find a
covering $\{T_i \to T\}$ of $\Sch_\proetale$ such that $s$ lifts
to a section of $\mathcal{F}$ over $T_i$
(Sites, Definition \ref{sites-definition-sheaves-injective-surjective}).
By Lemma \ref{lemma-w-contractible-proetale-cover}
we can refine $\{T_i \to T\}$ by a Zariski
covering of the form $T = \coprod_{j = 1, \ldots, m} V_j$.
Hence we get $t_j \in \mathcal{F}(U_j)$ mapping to $s|_{U_j}$.
Since Zariski coverings are coverings in $\Sch_\proetale$
(Lemma \ref{lemma-etale-proetale}) we conclude that
$\mathcal{F}(T) = \prod \mathcal{F}(U_j)$.
Thus, taking $t = (t_1, \ldots, t_m) \in \mathcal{F}(T)$
is a section mapping to $s$.
\end{proof}
\begin{lemma}
\label{lemma-get-many-weakly-contractible}
Let $\Sch_\proetale$ be a big pro-\'etale site as in
Definition \ref{definition-big-proetale-site}.
For every object $T$ of $\Sch_\proetale$ there exists
a covering $\{T_i \to T\}$ in $\Sch_\proetale$
with each $T_i$ affine and the spectrum of a w-contractible
ring. In particular, $T_i$ is weakly contractible in $\Sch_\proetale$.
\end{lemma}
\begin{proof}
For those readers who do not care about set-theoretical issues
this lemma is a trivial consequence of
Lemma \ref{lemma-w-contractible-is-weakly-contractible} and
Proposition \ref{proposition-find-w-contractible}.
Here are the details.
Choose an affine open covering $T = \bigcup U_i$. Write $U_i = \Spec(A_i)$.
Choose faithfully flat, ind-\'etale ring maps $A_i \to D_i$
such that $D_i$ is w-contractible as in
Proposition \ref{proposition-find-w-contractible}.
The family of morphisms $\{\Spec(D_i) \to T\}$ is a
pro-\'etale covering.
If we can show that $\Spec(D_i)$ is isomorphic to an object, say $T_i$,
of $\Sch_\proetale$, then $\{T_i \to T\}$ will be combinatorially
equivalent to a covering of $\Sch_\proetale$ by the construction
of $\Sch_\proetale$ in Definition \ref{definition-big-proetale-site}
and more precisely the application of
Sets, Lemma \ref{sets-lemma-coverings-site} in the last step.
To prove $\Spec(D_i)$ is isomorphic to an object of
$\Sch_\proetale$, it suffices to prove that
$|D_i| \leq Bound(\text{Size}(T))$ by the construction
of $\Sch_\proetale$ in Definition \ref{definition-big-proetale-site}
and more precisely the application of
Sets, Lemma \ref{sets-lemma-construct-category} in step (3).
Since $|A_i| \leq \text{size}(U_i) \leq \text{size}(T)$
by Sets, Lemmas \ref{sets-lemma-bound-affine} and
\ref{sets-lemma-bound-finite-type} we get
$|D_i| \leq \kappa^{2^{2^{2^\kappa}}}$ where $\kappa = \text{size}(T)$
by Remark \ref{remark-size-w-contractible}.
Thus by our choice of the function $Bound$ in
Definition \ref{definition-big-proetale-site} we win.
\end{proof}
\begin{lemma}
\label{lemma-proetale-induced}
Let $\Sch_\proetale$ be a big pro-\'etale site as in
Definition \ref{definition-big-proetale-site}.
Let $T \in \Ob(\Sch_\proetale)$.
Let $\{T_i \to T\}_{i \in I}$ be an arbitrary pro-\'etale covering of $T$.
There exists a covering $\{U_j \to T\}_{j \in J}$ of $T$ in the site
$\Sch_\proetale$ which refines $\{T_i \to T\}_{i \in I}$.
\end{lemma}
\begin{proof}
Namely, we first let $\{V_k \to T\}$ be a covering as in
Lemma \ref{lemma-get-many-weakly-contractible}.
Then the pro-\'etale coverings $\{T_i \times_T V_k \to V_k\}$
can be refined by a finite disjoint open covering
$V_k = V_{k, 1} \amalg \ldots \amalg V_{k, n_k}$, see
Lemma \ref{lemma-w-contractible-proetale-cover}.
Then $\{V_{k, i} \to T\}$ is a covering of $\Sch_\proetale$
which refines $\{T_i \to T\}_{i \in I}$.
\end{proof}
\begin{definition}
\label{definition-big-small-proetale}
Let $S$ be a scheme. Let $\Sch_\proetale$ be a big pro-\'etale
site containing $S$.
\begin{enumerate}
\item The {\it big pro-\'etale site of $S$}, denoted
$(\Sch/S)_\proetale$, is the site $\Sch_\proetale/S$
introduced in Sites, Section \ref{sites-section-localize}.
\item The {\it small pro-\'etale site of $S$}, which we denote
$S_\proetale$, is the full subcategory of $(\Sch/S)_\proetale$
whose objects are those $U/S$ such that $U \to S$ is weakly \'etale.
A covering of $S_\proetale$ is any covering $\{U_i \to U\}$ of
$(\Sch/S)_\proetale$ with $U \in \Ob(S_\proetale)$.
\item The {\it big affine pro-\'etale site of $S$}, denoted
$(\textit{Aff}/S)_\proetale$, is the full subcategory of
$(\Sch/S)_\proetale$ whose objects are affine $U/S$.
A covering of $(\textit{Aff}/S)_\proetale$ is any covering
$\{U_i \to U\}$ of $(\Sch/S)_\proetale$ which is a
standard pro-\'etale covering.
\end{enumerate}
\end{definition}
\noindent
It is not completely clear that the small pro-\'etale site and
the big affine pro-\'etale site are sites. We check this now.
\begin{lemma}
\label{lemma-verify-site-proetale}
Let $S$ be a scheme. Let $\Sch_\proetale$ be a big pro-\'etale site
containing $S$. Both $S_\proetale$ and $(\textit{Aff}/S)_\proetale$ are sites.
\end{lemma}
\begin{proof}
Let us show that $S_\proetale$ 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 \ref{sites-definition-site}.
Since $(\Sch/S)_\proetale$ is a site, it suffices to prove
that given any covering $\{U_i \to U\}$ of $(\Sch/S)_\proetale$
with $U \in \Ob(S_\proetale)$ we also have $U_i \in \Ob(S_\proetale)$.
This follows from the definitions
as the composition of weakly \'etale morphisms is weakly \'etale.
\medskip\noindent
To show that $(\textit{Aff}/S)_\proetale$ is a site, reasoning as above,
it suffices to show that the collection of standard pro-\'etale coverings
of affines satisfies properties (1), (2) and (3) of
Sites, Definition \ref{sites-definition-site}.
This follows from Lemma \ref{lemma-recognize-proetale-covering}
and the corresponding result for standard fpqc coverings
(Topologies, Lemma \ref{topologies-lemma-fpqc-affine-axioms}).
\end{proof}
\begin{lemma}
\label{lemma-fibre-products-proetale}
Let $S$ be a scheme. Let $\Sch_\proetale$ be a big pro-\'etale
site containing $S$. Let $\Sch$ be the category of all schemes.
\begin{enumerate}
\item The categories $\Sch_\proetale$, $(\Sch/S)_\proetale$,
$S_\proetale$, and $(\textit{Aff}/S)_\proetale$ have fibre products
agreeing with fibre products in $\Sch$.
\item The categories $\Sch_\proetale$, $(\Sch/S)_\proetale$,
$S_\proetale$ have equalizers agreeing with equalizers in $\Sch$.
\item The categories $(\Sch/S)_\proetale$, and $S_\proetale$ both have
a final object, namely $S/S$.
\item The category $\Sch_\proetale$ has a final object agreeing
with the final object of $\Sch$, namely $\Spec(\mathbf{Z})$.
\end{enumerate}
\end{lemma}
\begin{proof}
The category $\Sch_\proetale$ contains $\Spec(\mathbf{Z})$ and
is closed under products and fibre products by construction, see
Sets, Lemma \ref{sets-lemma-what-is-in-it}.
Suppose we have $U \to S$, $V \to U$, $W \to U$ morphisms
of schemes with $U, V, W \in \Ob(\Sch_\proetale)$.
The fibre product $V \times_U W$ in $\Sch_\proetale$
is a fibre product in $\Sch$ 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 $(\Sch/S)_\proetale$.
This proves the result for $(\Sch/S)_\proetale$.
If $U \to S$, $V \to U$ and $W \to U$ are weakly \'etale then so is
$V \times_U W \to S$ (see
More on Morphisms, Section \ref{more-morphisms-section-weakly-etale})
and hence we get fibre products for $S_\proetale$.
If $U, V, W$ are affine, so is $V \times_U W$ and hence we
get fibre products for $(\textit{Aff}/S)_\proetale$.
\medskip\noindent
Let $a, b : U \to V$ be two morphisms in $\Sch_\proetale$.
In this case the equalizer of $a$ and $b$ (in the category of schemes) is
$$
V
\times_{\Delta_{V/\Spec(\mathbf{Z})}, V \times_{\Spec(\mathbf{Z})} V, (a, b)}
(U \times_{\Spec(\mathbf{Z})} U)
$$
which is an object of $\Sch_\proetale$ by what we saw above.
Thus $\Sch_\proetale$ 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 $(\Sch/S)_\proetale$ has equalizers. Moreover, if
$U$ and $V$ are weakly-\'etale over $S$, then so is the equalizer
above as a fibre product of schemes weakly \'etale over $S$.
Thus $S_\proetale$ has equalizers. The statements on final objects
is clear.
\end{proof}
\noindent
Next, we check that the big affine pro-\'etale site defines the same
topos as the big pro-\'etale site.
\begin{lemma}
\label{lemma-affine-big-site-proetale}
Let $S$ be a scheme. Let $\Sch_\proetale$ be a big pro-\'etale
site containing $S$.
The functor $(\textit{Aff}/S)_\proetale \to (\Sch/S)_\proetale$
is a special cocontinuous functor. Hence it induces an equivalence
of topoi from $\Sh((\textit{Aff}/S)_\proetale)$ to
$\Sh((\Sch/S)_\proetale)$.
\end{lemma}
\begin{proof}
The notion of a special cocontinuous functor is introduced in
Sites, Definition \ref{sites-definition-special-cocontinuous-functor}.
Thus we have to verify assumptions (1) -- (5) of
Sites, Lemma \ref{sites-lemma-equivalence}.
Denote the inclusion functor
$u : (\textit{Aff}/S)_\proetale \to (\Sch/S)_\proetale$.
Being cocontinuous just means that any pro-\'etale covering of
$T/S$, $T$ affine, can be refined by a standard pro-\'etale
covering of $T$. This is the content of
Lemma \ref{lemma-proetale-affine}.
Hence (1) holds. We see $u$ is continuous simply because a standard
pro-\'etale covering is a pro-\'etale 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.
\end{proof}
\begin{lemma}
\label{lemma-put-in-T}
Let $\Sch_\proetale$ be a big pro-\'etale site.
Let $f : T \to S$ be a morphism in $\Sch_\proetale$.
The functor $T_\proetale \to (\Sch/S)_\proetale$
is cocontinuous and induces a morphism of topoi
$$
i_f :
\Sh(T_\proetale)
\longrightarrow
\Sh((\Sch/S)_\proetale)
$$
For a sheaf $\mathcal{G}$ on $(\Sch/S)_\proetale$
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.
\end{lemma}
\begin{proof}
Denote the functor $u : T_\proetale \to (\Sch/S)_\proetale$.
In other words, given a weakly \'etale morphism $j : U \to T$ corresponding
to an object of $T_\proetale$ we set $u(U \to T) = (f \circ j : U \to S)$.
This functor commutes with fibre products, see
Lemma \ref{lemma-fibre-products-proetale}.
Moreover, $T_\proetale$ has equalizers and $u$ commutes with them
by Lemma \ref{lemma-fibre-products-proetale}.
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 \ref{sites-lemma-when-shriek}
and \ref{sites-lemma-preserve-equalizers}.
\end{proof}
\begin{lemma}
\label{lemma-at-the-bottom}
Let $S$ be a scheme. Let $\Sch_\proetale$ be a big pro-\'etale
site containing $S$.
The inclusion functor $S_\proetale \to (\Sch/S)_\proetale$
satisfies the hypotheses of Sites, Lemma \ref{sites-lemma-bigger-site}
and hence induces a morphism of sites
$$
\pi_S : (\Sch/S)_\proetale \longrightarrow S_\proetale
$$
and a morphism of topoi
$$
i_S : \Sh(S_\proetale) \longrightarrow \Sh((\Sch/S)_\proetale)
$$
such that $\pi_S \circ i_S = \text{id}$. Moreover, $i_S = i_{\text{id}_S}$
with $i_{\text{id}_S}$ as in Lemma \ref{lemma-put-in-T}. In particular the
functor $i_S^{-1} = \pi_{S, *}$ is described by the rule
$i_S^{-1}(\mathcal{G})(U/S) = \mathcal{G}(U/S)$.
\end{lemma}
\begin{proof}
In this case the functor $u : S_\proetale \to (\Sch/S)_\proetale$,
in addition to the properties seen in the proof of
Lemma \ref{lemma-put-in-T} above, also is fully faithful
and transforms the final object into the final object.
The lemma follows from Sites, Lemma \ref{sites-lemma-bigger-site}.
\end{proof}
\begin{definition}
\label{definition-restriction-small-proetale}
In the situation of
Lemma \ref{lemma-at-the-bottom}
the functor $i_S^{-1} = \pi_{S, *}$ is often
called the {\it restriction to the small pro-\'etale site}, and for a sheaf
$\mathcal{F}$ on the big pro-\'etale site we denote
$\mathcal{F}|_{S_\proetale}$ this restriction.
\end{definition}
\noindent
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*}
\Mor_{\Sh(S_\proetale)}(\mathcal{F}|_{S_\proetale}, \mathcal{G})
& =
\Mor_{\Sh((\Sch/S)_\proetale)}(\mathcal{F},
i_{S, *}\mathcal{G}) \\
\Mor_{\Sh(S_\proetale)}(\mathcal{G}, \mathcal{F}|_{S_\proetale})
& =
\Mor_{\Sh((\Sch/S)_\proetale)}(\pi_S^{-1}\mathcal{G}, \mathcal{F})
\end{align*}
Moreover, we have $(i_{S, *}\mathcal{G})|_{S_\proetale} = \mathcal{G}$
and we have $(\pi_S^{-1}\mathcal{G})|_{S_\proetale} = \mathcal{G}$.
\begin{lemma}
\label{lemma-morphism-big}
Let $\Sch_\proetale$ be a big pro-\'etale site.
Let $f : T \to S$ be a morphism in $\Sch_\proetale$.
The functor
$$
u : (\Sch/T)_\proetale \longrightarrow (\Sch/S)_\proetale, \quad
V/T \longmapsto V/S
$$
is cocontinuous, and has a continuous right adjoint
$$
v : (\Sch/S)_\proetale \longrightarrow (\Sch/T)_\proetale, \quad
(U \to S) \longmapsto (U \times_S T \to T).
$$
They induce the same morphism of topoi
$$
f_{big} :
\Sh((\Sch/T)_\proetale)
\longrightarrow
\Sh((\Sch/S)_\proetale)
$$
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.
\end{lemma}
\begin{proof}
The functor $u$ is cocontinuous, continuous, and commutes with fibre products
and equalizers (details omitted; compare with proof of
Lemma \ref{lemma-put-in-T}). Hence
Sites, Lemmas \ref{sites-lemma-when-shriek} and
\ref{sites-lemma-preserve-equalizers}
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 $\Mor_S(u(U), V) = \Mor_T(U, V \times_S T)$
as desired. Thus we may apply
Sites, Lemmas \ref{sites-lemma-have-functor-other-way} and
\ref{sites-lemma-have-functor-other-way-morphism}
to get the formula for $f_{big, *}$.
\end{proof}
\begin{lemma}
\label{lemma-morphism-big-small}
Let $\Sch_\proetale$ be a big pro-\'etale site.
Let $f : T \to S$ be a morphism in $\Sch_\proetale$.
\begin{enumerate}
\item We have $i_f = f_{big} \circ i_T$ with $i_f$ as in
Lemma \ref{lemma-put-in-T} and $i_T$ as in
Lemma \ref{lemma-at-the-bottom}.
\item The functor $S_\proetale \to T_\proetale$,
$(U \to S) \mapsto (U \times_S T \to T)$ is continuous and induces
a morphism of topoi
$$
f_{small} : \Sh(T_\proetale) \longrightarrow \Sh(S_\proetale).
$$
We have $f_{small, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times_S T/T)$.
\item We have a commutative diagram of morphisms of sites
$$
\xymatrix{
T_\proetale \ar[d]_{f_{small}} &
(\Sch/T)_\proetale \ar[d]^{f_{big}} \ar[l]^{\pi_T}\\
S_\proetale &
(\Sch/S)_\proetale \ar[l]_{\pi_S}
}
$$
so that $f_{small} \circ \pi_T = \pi_S \circ f_{big}$ as morphisms of topoi.
\item We have $f_{small} = \pi_S \circ f_{big} \circ i_T = \pi_S \circ i_f$.
\end{enumerate}
\end{lemma}
\begin{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).
\medskip\noindent
The functor $u : S_\proetale \to T_\proetale$,
$u(U \to S) = (U \times_S T \to T)$
transforms coverings into coverings and commutes with fibre products,
see Lemmas \ref{lemma-proetale} and \ref{lemma-fibre-products-proetale}.
Moreover, both $S_\proetale$, $T_\proetale$ have final objects,
namely $S/S$ and $T/T$ and $u(S/S) = T/T$. Hence by
Sites, Proposition \ref{sites-proposition-get-morphism}
the functor $u$ corresponds to a morphism of sites
$T_\proetale \to S_\proetale$. This in turn gives rise to the
morphism of topoi, see
Sites, Lemma \ref{sites-lemma-morphism-sites-topoi}. The description
of the pushforward is clear from these references.
\medskip\noindent
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 $U \mapsto U \times_S T$.
\medskip\noindent
Statement (4) follows from (3) by precomposing with $i_T$.
\end{proof}
\noindent
In the situation of the lemma, using the terminology of
Definition \ref{definition-restriction-small-proetale}
we have: for $\mathcal{F}$ a sheaf on the big pro-\'etale site of $T$
$$
(f_{big, *}\mathcal{F})|_{S_\proetale} =
f_{small, *}(\mathcal{F}|_{T_\proetale}),
$$
This equality is clear from the commutativity of the diagram of
sites of the lemma, since restriction to the small pro-\'etale site of
$T$, resp.\ $S$ is given by $\pi_{T, *}$, resp.\ $\pi_{S, *}$. A similar
formula involving pullbacks and restrictions is false.
\begin{lemma}
\label{lemma-composition-proetale}
Given schemes $X$, $Y$, $Y$ in $\Sch_\proetale$
and morphisms $f : X \to Y$, $g : Y \to Z$ we have
$g_{big} \circ f_{big} = (g \circ f)_{big}$ and
$g_{small} \circ f_{small} = (g \circ f)_{small}$.
\end{lemma}
\begin{proof}
This follows from the simple description of pushforward
and pullback for the functors on the big sites from
Lemma \ref{lemma-morphism-big}. For the functors
on the small sites this follows from the description of
the pushforward functors in Lemma \ref{lemma-morphism-big-small}.
\end{proof}
\noindent
We can think about a sheaf on the big pro-\'etale site of $S$ as a collection
of sheaves on the small pro-\'etale site on schemes over $S$.
\begin{lemma}
\label{lemma-characterize-sheaf-big}
Let $S$ be a scheme contained in a big pro-\'etale site $\Sch_\proetale$.
A sheaf $\mathcal{F}$ on the big pro-\'etale site $(\Sch/S)_\proetale$
is given by the following data:
\begin{enumerate}
\item for every $T/S \in \Ob((\Sch/S)_\proetale)$ a sheaf
$\mathcal{F}_T$ on $T_\proetale$,
\item for every $f : T' \to T$ in
$(\Sch/S)_\proetale$ a map
$c_f : f_{small}^{-1}\mathcal{F}_T \to \mathcal{F}_{T'}$.
\end{enumerate}
These data are subject to the following conditions:
\begin{enumerate}
\item[(a)] given any $f : T' \to T$ and $g : T'' \to T'$ in
$(\Sch/S)_\proetale$ the composition
$g_{small}^{-1}c_f \circ c_g$ is equal to $c_{f \circ g}$, and
\item[(b)] if $f : T' \to T$ in $(\Sch/S)_\proetale$
is weakly \'etale then $c_f$ is an isomorphism.
\end{enumerate}
\end{lemma}
\begin{proof}
Identical to the proof of
Topologies, Lemma \ref{topologies-lemma-characterize-sheaf-big-etale}.
\end{proof}
\begin{lemma}
\label{lemma-alternative}
Let $S$ be a scheme. Let $S_{affine, \proetale}$ denote the full subcategory
of $S_\proetale$ consisting of affine objects. A covering of
$S_{affine, \proetale}$ will be a standard \'etale covering, see
Definition \ref{definition-standard-proetale}.
Then restriction
$$
\mathcal{F} \longmapsto \mathcal{F}|_{S_{affine, \etale}}
$$
defines an equivalence of topoi
$\Sh(S_\proetale) \cong \Sh(S_{affine, \proetale})$.
\end{lemma}
\begin{proof}
This you can show directly from the definitions, and is a good exercise.
But it also follows immediately from
Sites, Lemma \ref{sites-lemma-equivalence}
by checking that the inclusion functor
$S_{affine, \proetale} \to S_\proetale$
is a special cocontinuous functor (see
Sites, Definition \ref{sites-definition-special-cocontinuous-functor}).
\end{proof}
\begin{lemma}
\label{lemma-affine-alternative}
Let $S$ be an affine scheme. Let $S_{app}$ denote the full subcategory
of $S_\proetale$ consisting of affine objects $U$ such that
$\mathcal{O}(S) \to \mathcal{O}(U)$ is ind-\'etale. A covering of
$S_{app}$ will be a standard pro-\'etale covering, see
Definition \ref{definition-standard-proetale}.
Then restriction
$$
\mathcal{F} \longmapsto \mathcal{F}|_{S_{app}}
$$
defines an equivalence of topoi $\Sh(S_\proetale) \cong \Sh(S_{app})$.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-alternative} we may replace $S_\proetale$ by
$S_{affine, \proetale}$.
The lemma follows from Sites, Lemma \ref{sites-lemma-equivalence}
by checking that the inclusion functor $S_{app} \to S_{affine, \proetale}$
is a special cocontinuous functor, see
Sites, Definition \ref{sites-definition-special-cocontinuous-functor}.
The conditions of Sites, Lemma \ref{sites-lemma-equivalence}
follow immediately from the definition and the facts
(a) any object $U$ of $S_{affine, \proetale}$ has a covering
$\{V \to U\}$ with $V$ ind-\'etale over $X$
(Proposition \ref{proposition-weakly-etale})
and (b) the functor $u$ is fully faithful.
\end{proof}
\noindent
Next we show that cohomology of sheaves is independent of the choice
of a partial universe. Namely, the functor $g_*$ of the lemma below
is an embedding of pro-\'etale topoi which does not change cohomology.
\begin{lemma}
\label{lemma-proetale-cohomology-independent-partial-universe}
Let $S$ be a scheme. Let $S_\proetale \subset S_\proetale'$ be
two small pro-\'etale sites of $S$ as constructed in
Definition \ref{definition-big-small-proetale}. Then the inclusion functor
satisfies the assumptions of
Sites, Lemma \ref{sites-lemma-bigger-site}.
Hence there exist morphisms of topoi
$$
\xymatrix{
\Sh(S_\proetale) \ar[r]^g &
\Sh(S_\proetale') \ar[r]^f &
\Sh(S_\proetale)
}
$$
whose composition is isomorphic to the identity and with $f_* = g^{-1}$.
Moreover,
\begin{enumerate}
\item for $\mathcal{F}' \in \textit{Ab}(S_\proetale')$ we have
$H^p(S_\proetale', \mathcal{F}') = H^p(S_\proetale, g^{-1}\mathcal{F}')$,
\item for $\mathcal{F} \in \textit{Ab}(S_\proetale)$ we have
$$
H^p(S_\proetale, \mathcal{F}) =
H^p(S_\proetale', g_*\mathcal{F}) =
H^p(S_\proetale', f^{-1}\mathcal{F}).
$$
\end{enumerate}
\end{lemma}
\begin{proof}
The inclusion functor is fully faithful and continuous.
We have seen that $S_\proetale$ and $S_\proetale'$ have fibre products
and final objects and that our functor commutes with these
(Lemma \ref{lemma-fibre-products-proetale}).
It follows from Lemma \ref{lemma-proetale-induced}
that the inclusion functor is cocontinuous.
Hence the existence of $f$ and $g$ follows from
Sites, Lemma \ref{sites-lemma-bigger-site}.
The equality in (1) is
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-bigger-site}.
Part (2) follows from (1) as
$\mathcal{F} = g^{-1}g_*\mathcal{F} = g^{-1}f^{-1}\mathcal{F}$.
\end{proof}
\begin{lemma}
\label{lemma-proetale-subcanonical}
Let $S$ be a scheme. The topology on each of the pro-\'etale sites
$S_\proetale$, $(\Sch/S)_\proetale$, $S_{affine, \proetale}$, and
$(\textit{Aff}/S)_\proetale$ is subcanonical.
\end{lemma}
\begin{proof}
Combine Lemma \ref{lemma-recognize-proetale-covering} and
Descent, Lemma \ref{descent-lemma-fpqc-universal-effective-epimorphisms}.
\end{proof}
\begin{lemma}
\label{lemma-proetale-enough-w-contractible}
Let $S$ be a scheme. The pro-\'etale sites
$S_\proetale$, $(\Sch/S)_\proetale$, $S_{affine, \proetale}$, and
$(\textit{Aff}/S)_\proetale$ and if $S$ is affine $S_{app}$
have enough quasi-compact, weakly contractible
objects, see Sites, Definition \ref{sites-definition-w-contractible}.
\end{lemma}
\begin{proof}
Follows immediately from Lemma \ref{lemma-get-many-weakly-contractible}.
\end{proof}
```

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