Lemma 61.25.2 (09BL)—The Stacks project

Lemma 61.25.2. Let $Z \to X$ be a closed immersion morphism of affine schemes. The corresponding morphism of topoi $i = i_{pro\text{-}\acute{e}tale}$ is equal to the morphism of topoi associated to the fully faithful cocontinuous functor $v : Z_{app} \to X_{app}$ of Lemma 61.25.1. It follows that

$i^{-1}\mathcal{F}$ is the sheaf associated to the presheaf $V \mapsto \mathcal{F}(v(V))$ ,
for a weakly contractible object $V$ of $Z_{app}$ we have $i^{-1}\mathcal{F}(V) = \mathcal{F}(v(V))$ ,
$i^{-1} : \mathop{\mathit{Sh}}\nolimits (X_{pro\text{-}\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Z_{pro\text{-}\acute{e}tale})$ has a left adjoint $i^{Sh}_!$ ,
$i^{-1} : \textit{Ab}(X_{pro\text{-}\acute{e}tale}) \to \textit{Ab}(Z_{pro\text{-}\acute{e}tale})$ has a left adjoint $i_!$ ,
$\text{id} \to i^{-1}i^{Sh}_!$ , $\text{id} \to i^{-1}i_!$ , and $i^{-1}i_* \to \text{id}$ are isomorphisms, and
$i_*$ , $i^{Sh}_!$ and $i_!$ are fully faithful.

Proof. By Lemma 61.12.21 we may describe $i_{pro\text{-}\acute{e}tale}$ in terms of the morphism of sites $u : X_{app} \to Z_{app}$ , $V \mapsto V \times _ X Z$ . The first statement of the lemma follows from Sites, Lemmas 7.22.1 and 7.22.2 (but with the roles of $u$ and $v$ reversed).

Proof of (1). By the description of $i$ as the morphism of topoi associated to $v$ this holds by the construction, see Sites, Lemma 7.21.1.

Proof of (2). Since the functor $v$ sends coverings to coverings by Lemma 61.25.1 we see that the presheaf $\mathcal{G} : V \mapsto \mathcal{F}(v(V))$ is a separated presheaf (Sites, Definition 7.10.9). Hence the sheafification of $\mathcal{G}$ is $\mathcal{G}^+$ , see Sites, Theorem 7.10.10. Next, let $V$ be a weakly contractible object of $Z_{app}$ . Let $\mathcal{V} = \{ V_ i \to V\} _{i = 1, \ldots , n}$ be any covering in $Z_{app}$ . Set $\mathcal{V}' = \{ \coprod V_ i \to V\}$ . Since $v$ commutes with finite disjoint unions (as a left adjoint or by the construction) and since $\mathcal{F}$ sends finite disjoint unions into products, we see that

$H^0(\mathcal{V}, \mathcal{G}) = H^0(\mathcal{V}', \mathcal{G})$

(notation as in Sites, Section 7.10; compare with Étale Cohomology, Lemma 59.22.1). Thus we may assume the covering is given by a single morphism, like so $\{ V' \to V\}$ . Since $V$ is weakly contractible, this covering can be refined by the trivial covering $\{ V \to V\}$ . It therefore follows that the value of $\mathcal{G}^+ = i^{-1}\mathcal{F}$ on $V$ is simply $\mathcal{F}(v(V))$ and (2) is proved.

Proof of (3). Every object of $Z_{app}$ has a covering by weakly contractible objects (Lemma 61.13.4). By the above we see that we would have $i^{Sh}_!h_ V = h_{v(V)}$ for $V$ weakly contractible if $i^{Sh}_!$ existed. The existence of $i^{Sh}_!$ then follows from Sites, Lemma 7.24.1.

Proof of (4). Existence of $i_!$ follows in the same way by setting $i_!\mathbf{Z}_ V = \mathbf{Z}_{v(V)}$ for $V$ weakly contractible in $Z_{app}$ , using similar for direct sums, and applying Homology, Lemma 12.29.6. Details omitted.

Proof of (5). Let $V$ be a contractible object of $Z_{app}$ . Then $i^{-1}i^{Sh}_!h_ V = i^{-1}h_{v(V)} = h_{u(v(V))} = h_ V$ . (It is a general fact that $i^{-1}h_ U = h_{u(U)}$ .) Since the sheaves $h_ V$ for $V$ contractible generate $\mathop{\mathit{Sh}}\nolimits (Z_{app})$ (Sites, Lemma 7.12.5) we conclude $\text{id} \to i^{-1}i^{Sh}_!$ is an isomorphism. Similarly for the map $\text{id} \to i^{-1}i_!$ . Then $(i^{-1}i_*\mathcal{H})(V) = i_*\mathcal{H}(v(V)) = \mathcal{H}(u(v(V))) = \mathcal{H}(V)$ and we find that $i^{-1}i_* \to \text{id}$ is an isomorphism.

The fully faithfulness statements of (6) now follow from Categories, Lemma 4.24.4. $\square$

The Stacks project

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