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

  1. $i^{-1}\mathcal{F}$ is the sheaf associated to the presheaf $V \mapsto \mathcal{F}(v(V))$,

  2. for a weakly contractible object $V$ of $Z_{app}$ we have $i^{-1}\mathcal{F}(V) = \mathcal{F}(v(V))$,

  3. $i^{-1} : \mathop{\mathit{Sh}}\nolimits (X_{pro\text{-}\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Z_{pro\text{-}\acute{e}tale})$ has a left adjoint $i^{Sh}_!$,

  4. $i^{-1} : \textit{Ab}(X_{pro\text{-}\acute{e}tale}) \to \textit{Ab}(Z_{pro\text{-}\acute{e}tale})$ has a left adjoint $i_!$,

  5. $\text{id} \to i^{-1}i^{Sh}_!$, $\text{id} \to i^{-1}i_!$, and $i^{-1}i_* \to \text{id}$ are isomorphisms, and

  6. $i_*$, $i^{Sh}_!$ 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, Lemma 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. $\square$


Comments (0)


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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09BL. Beware of the difference between the letter 'O' and the digit '0'.