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61.31 Change of partial universe

We advise the reader to skip this section: here we show that cohomology of sheaves in the pro-étale topology is independent of the choice of partial universe. Namely, the functor $g_*$ of Lemma 61.31.2 below is an embedding of small pro-étale topoi which does not change cohomology. For big pro-étale sites we have Lemmas 61.31.3 and 61.31.4 saying essentially the same thing.

But first, as promised in Section 61.12 we prove 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.

Lemma 61.31.1. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site as in Definition 61.12.7. Let $T \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{pro\text{-}\acute{e}tale})$. 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 $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ which refines $\{ T_ i \to T\} _{i \in I}$.

Proof. Namely, we first let $\{ V_ k \to T\} $ be a covering as in Lemma 61.13.3. 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 61.13.1. Then $\{ V_{k, i} \to T\} $ is a covering of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ which refines $\{ T_ i \to T\} _{i \in I}$. $\square$

We first state and prove the comparison for the small pro-étale sites. Note that we are not claiming that the small pro-étale topos of a scheme is independent of the choice of partial universe; this isn't true in contrast with the case of the small étale topos (Étale Cohomology, Lemma 59.21.3).

Lemma 61.31.2. 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 61.12.8. Then the inclusion functor satisfies the assumptions of Sites, Lemma 7.21.8. Hence there exist morphisms of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale}) \ar[r]^ g & \mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale}') \ar[r]^ f & \mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale}) } \]

whose composition is isomorphic to the identity and with $f_* = g^{-1}$. Moreover,

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

  2. 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 61.12.10). It follows from Lemma 61.31.1 that the inclusion functor is cocontinuous. Hence the existence of $f$ and $g$ follows from Sites, Lemma 7.21.8. The equality in (1) is Cohomology on Sites, Lemma 21.7.2. Part (2) follows from (1) as $\mathcal{F} = g^{-1}g_*\mathcal{F} = g^{-1}f^{-1}\mathcal{F}$. $\square$

Next, we prove a corresponding result for the big pro-étale topoi.

Lemma 61.31.3. Suppose given big sites $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ and $\mathit{Sch}'_{pro\text{-}\acute{e}tale}$ as in Definition 61.12.7. Assume that $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ is contained in $\mathit{Sch}'_{pro\text{-}\acute{e}tale}$. The inclusion functor $\mathit{Sch}_{pro\text{-}\acute{e}tale}\to \mathit{Sch}'_{pro\text{-}\acute{e}tale}$ satisfies the assumptions of Sites, Lemma 7.21.8. There are morphisms of topoi

\begin{eqnarray*} g : \mathop{\mathit{Sh}}\nolimits (\mathit{Sch}_{pro\text{-}\acute{e}tale}) & \longrightarrow & \mathop{\mathit{Sh}}\nolimits (\mathit{Sch}'_{pro\text{-}\acute{e}tale}) \\ f : \mathop{\mathit{Sh}}\nolimits (\mathit{Sch}'_{pro\text{-}\acute{e}tale}) & \longrightarrow & \mathop{\mathit{Sh}}\nolimits (\mathit{Sch}_{pro\text{-}\acute{e}tale}) \end{eqnarray*}

such that $f \circ g \cong \text{id}$. For any object $S$ of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ the inclusion functor $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}'/S)_{pro\text{-}\acute{e}tale}$ satisfies the assumptions of Sites, Lemma 7.21.8 also. Hence similarly we obtain morphisms

\begin{eqnarray*} g : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}) & \longrightarrow & \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}'/S)_{pro\text{-}\acute{e}tale}) \\ f : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}'/S)_{pro\text{-}\acute{e}tale}) & \longrightarrow & \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}) \end{eqnarray*}

with $f \circ g \cong \text{id}$.

Proof. Assumptions (b), (c), and (e) of Sites, Lemma 7.21.8 are immediate for the functors $\mathit{Sch}_{pro\text{-}\acute{e}tale}\to \mathit{Sch}'_{pro\text{-}\acute{e}tale}$ and $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}'/S)_{pro\text{-}\acute{e}tale}$. Property (a) holds by Lemma 61.31.1. Property (d) holds because fibre products in the categories $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, $\mathit{Sch}'_{pro\text{-}\acute{e}tale}$ exist and are compatible with fibre products in the category of schemes. $\square$

Lemma 61.31.4. Let $S$ be a scheme. Let $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ and $(\mathit{Sch}'/S)_{pro\text{-}\acute{e}tale}$ be two big pro-étale sites of $S$ as in Definition 61.12.8. Assume that the first is contained in the second. In this case

  1. for any abelian sheaf $\mathcal{F}'$ defined on $(\mathit{Sch}'/S)_{pro\text{-}\acute{e}tale}$ and any object $U$ of $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ we have

    \[ H^ p(U, \mathcal{F}'|_{(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}}) = H^ p(U, \mathcal{F}') \]

    In words: the cohomology of $\mathcal{F}'$ over $U$ computed in the bigger site agrees with the cohomology of $\mathcal{F}'$ restricted to the smaller site over $U$.

  2. for any abelian sheaf $\mathcal{F}$ on $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ there is an abelian sheaf $\mathcal{F}'$ on $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}'$ whose restriction to $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is isomorphic to $\mathcal{F}$.

Proof. By Lemma 61.31.3 the inclusion functor $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}'/S)_{pro\text{-}\acute{e}tale}$ satisfies the assumptions of Sites, Lemma 7.21.8. This implies (2) and (1) follows from Cohomology on Sites, Lemma 21.7.2. $\square$


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