Lemma 61.16.1. Let $X$ be a scheme. Let $\mathcal{F}$ be a presheaf of sets on $X_{pro\text{-}\acute{e}tale}$ which sends finite disjoint unions to products. Then $\mathcal{F}^\# (W) = \mathcal{F}(W)$ if $W$ is an affine weakly contractible object of $X_{pro\text{-}\acute{e}tale}$.

## 61.16 Comparing topologies

This section is the analogue of Étale Cohomology, Section 59.39.

**Proof.**
Recall that $\mathcal{F}^\# $ is equal to $(\mathcal{F}^+)^+$, see Sites, Theorem 7.10.10, where $\mathcal{F}^+$ is the presheaf which sends an object $U$ of $X_{pro\text{-}\acute{e}tale}$ to $\mathop{\mathrm{colim}}\nolimits H^0(\mathcal{U}, \mathcal{F})$ where the colimit is over all pro-étale coverings $\mathcal{U}$ of $U$. Thus it suffices to prove that (a) $\mathcal{F}^+$ sends finite disjoint unions to products and (b) sends $W$ to $\mathcal{F}(W)$. If $U = U_1 \amalg U_2$, then given a pro-étale covering $\mathcal{U} = \{ f_ j : V_ j \to U\} $ of $U$ we obtain pro-étale coverings $\mathcal{U}_ i = \{ f_ j^{-1}(U_ i) \to U_ i\} $ and we clearly have

because $\mathcal{F}$ sends finite disjoint unions to products (this includes the condition that $\mathcal{F}$ sends the empty scheme to the singleton). This proves (a). Finally, any pro-étale covering of $W$ can be refined by a finite disjoint union decomposition $W = W_1 \amalg \ldots W_ n$ by Lemma 61.13.2. Hence $\mathcal{F}^+(W) = \mathcal{F}(W)$ exactly because the value of $\mathcal{F}$ on $W$ is the product of the values of $\mathcal{F}$ on the $W_ j$. This proves (b). $\square$

Lemma 61.16.2. Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a sheaf of sets on $X_{pro\text{-}\acute{e}tale}$. If $W$ is an affine weakly contractible object of $X_{pro\text{-}\acute{e}tale}$, then

where the colimit is over morphisms $W \to V$ over $Y$ with $V \in Y_{pro\text{-}\acute{e}tale}$.

**Proof.**
Recall that $f_{small}^{-1}\mathcal{F}$ is the sheaf associated to the presheaf

on $X_{\acute{e}tale}$, see Sites, Sections 7.14 and 7.13; we've surpressed from the notation that the colimit is over the opposite of the category $\{ U \to V, V \in Y_{pro\text{-}\acute{e}tale}\} $. By Lemma 61.16.1 it suffices to prove that $u_ p\mathcal{F}$ sends finite disjoint unions to products. Suppose that $U = U_1 \amalg U_2$ is a disjoint union of open and closed subschemes. There is a functor

which is initial (Categories, Definition 4.17.3). Hence the corresponding functor on opposite categories is cofinal and by Categories, Lemma 4.17.2 we see that $u_ p\mathcal{F}$ on $U$ is the colimit of the values $\mathcal{F}(V_1 \amalg V_2)$ over the product category. Since $\mathcal{F}$ is a sheaf it sends disjoint unions to products and we conclude $u_ p\mathcal{F}$ does too. $\square$

Lemma 61.16.3. Let $S$ be a scheme. Consider the morphism

of Lemma 61.12.13. Let $\mathcal{F}$ be a sheaf on $S_{pro\text{-}\acute{e}tale}$. Then $\pi _ S^{-1}\mathcal{F}$ is given by the rule

where $f : T \to S$. Moreover, $\pi _ S^{-1}\mathcal{F}$ satisfies the sheaf condition with respect to fpqc coverings.

**Proof.**
Observe that we have a morphism $i_ f : \mathop{\mathit{Sh}}\nolimits (T_{pro\text{-}\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale})$ such that $\pi _ S \circ i_ f = f_{small}$ as morphisms $T_{pro\text{-}\acute{e}tale}\to S_{pro\text{-}\acute{e}tale}$, see Lemma 61.12.12. Since pullback is transitive we see that $i_ f^{-1} \pi _ S^{-1}\mathcal{F} = f_{small}^{-1}\mathcal{F}$ as desired.

Let $\{ g_ i : T_ i \to T\} _{i \in I}$ be an fpqc covering. The final statement means the following: Given a sheaf $\mathcal{G}$ on $T_{pro\text{-}\acute{e}tale}$ and given sections $s_ i \in \Gamma (T_ i, g_{i, small}^{-1}\mathcal{G})$ whose pullbacks to $T_ i \times _ T T_ j$ agree, there is a unique section $s$ of $\mathcal{G}$ over $T$ whose pullback to $T_ i$ agrees with $s_ i$. We will prove this statement when $T$ is affine and the covering is given by a single surjective flat morphism $T' \to T$ of affines and omit the reduction of the general case to this case.

Let $g : T' \to T$ be a surjective flat morphism of affines and let $s' \in g_{small}^{-1}\mathcal{G}(T')$ be a section with $\text{pr}_0^*s' = \text{pr}_1^*s'$ on $T' \times _ T T'$. Choose a surjective weakly étale morphism $W \to T'$ with $W$ affine and weakly contractible, see Lemma 61.13.5. By Lemma 61.16.2 the restriction $s'|_ W$ is an element of $\mathop{\mathrm{colim}}\nolimits _{W \to U} \mathcal{G}(U)$. Choose $\phi : W \to U_0$ and $s_0 \in \mathcal{G}(U_0)$ corresponding to $s'$. Choose a surjective weakly étale morphism $V \to W \times _ T W$ with $V$ affine and weakly contractible. Denote $a, b : V \to W$ the induced morphisms. Since $a^*(s'|_ W) = b^*(s'|_ W)$ and since the category $\{ V \to U, U \in T_{pro\text{-}\acute{e}tale}\} $ is cofiltered (this is clear but see Sites, Lemma 7.14.6 if in doubt), we see that the two morphisms $\phi \circ a , \phi \circ b : V \to U_0$ have to be equal. By the results in Descent, Section 35.12 (especially Descent, Lemma 35.12.7) it follows there is a unique morphism $T \to U_0$ such that $\phi $ is the composition of this morphism with the structure morphism $W \to T$ (small detail omitted). Then we can let $s$ be the pullback of $s_0$ by this morphism. We omit the verification that $s$ pulls back to $s'$ on $T'$. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)