The Stacks project

61.25 Closed immersions and pro-étale sites

It is not clear (and likely false) that a closed immersion of schemes determines an exact pushforward on abelian pro-étale sheaves.

Lemma 61.25.1. Let $i : Z \to X$ be a closed immersion morphism of affine schemes. Denote $X_{app}$ and $Z_{app}$ the sites introduced in Lemma 61.12.21. The base change functor

\[ u : X_{app} \to Z_{app},\quad U \longmapsto u(U) = U \times _ X Z \]

is continuous and has a fully faithful left adjoint $v$. For $V$ in $Z_{app}$ the morphism $V \to v(V)$ is a closed immersion identifying $V$ with $u(v(V)) = v(V) \times _ X Z$ and every point of $v(V)$ specializes to a point of $V$. The functor $v$ is cocontinuous and sends coverings to coverings.

Proof. The existence of the adjoint follows immediately from Lemma 61.7.7 and the definitions. It is clear that $u$ is continuous from the definition of coverings in $X_{app}$.

Write $X = \mathop{\mathrm{Spec}}(A)$ and $Z = \mathop{\mathrm{Spec}}(A/I)$. Let $V = \mathop{\mathrm{Spec}}(\overline{C})$ be an object of $Z_{app}$ and let $v(V) = \mathop{\mathrm{Spec}}(C)$. We have seen in the statement of Lemma 61.7.7 that $V$ equals $v(V) \times _ X Z = \mathop{\mathrm{Spec}}(C/IC)$. Any $g \in C$ which maps to an invertible element of $C/IC = \overline{C}$ is invertible in $C$. Namely, we have the $A$-algebra maps $C \to C_ g \to C/IC$ and by adjointness we obtain an $C$-algebra map $C_ g \to C$. Thus every point of $v(V)$ specializes to a point of $V$.

Suppose that $\{ V_ i \to V\} $ is a covering in $Z_{app}$. Then $\{ v(V_ i) \to v(V)\} $ is a finite family of morphisms of $Z_{app}$ such that every point of $V \subset v(V)$ is in the image of one of the maps $v(V_ i) \to v(V)$. As the morphisms $v(V_ i) \to v(V)$ are flat (since they are weakly étale) we conclude that $\{ v(V_ i) \to v(V)\} $ is jointly surjective. This proves that $v$ sends coverings to coverings.

Let $V$ be an object of $Z_{app}$ and let $\{ U_ i \to v(V)\} $ be a covering in $X_{app}$. Then we see that $\{ u(U_ i) \to u(v(V)) = V\} $ is a covering of $Z_{app}$. By adjointness we obtain morphisms $v(u(U_ i)) \to U_ i$. Thus the family $\{ v(u(U_ i)) \to v(V)\} $ refines the given covering and we conclude that $v$ is cocontinuous. $\square$

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, 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. $\square$

Lemma 61.25.3. Let $i : Z \to X$ be a closed immersion of schemes. Then

  1. $i_{pro\text{-}\acute{e}tale}^{-1}$ commutes with limits,

  2. $i_{{pro\text{-}\acute{e}tale}, *}$ is fully faithful, and

  3. $i_{pro\text{-}\acute{e}tale}^{-1}i_{{pro\text{-}\acute{e}tale}, *} \cong \text{id}_{\mathop{\mathit{Sh}}\nolimits (Z_{pro\text{-}\acute{e}tale})}$.

Proof. Assertions (2) and (3) are equivalent by Sites, Lemma 7.41.1. Parts (1) and (3) are (Zariski) local on $X$, hence we may assume that $X$ is affine. In this case the result follows from Lemma 61.25.2. $\square$

Lemma 61.25.4. Let $i : Z \to X$ be an integral universally injective and surjective morphism of schemes. Then $i_{{pro\text{-}\acute{e}tale}, *}$ and $i_{pro\text{-}\acute{e}tale}^{-1}$ are quasi-inverse equivalences of categories of pro-étale topoi.

Proof. There is an immediate reduction to the case that $X$ is affine. Then $Z$ is affine too. Set $A = \mathcal{O}(X)$ and $B = \mathcal{O}(Z)$. Then the categories of étale algebras over $A$ and $B$ are equivalent, see Étale Cohomology, Theorem 59.45.2 and Remark 59.45.3. Thus the categories of ind-étale algebras over $A$ and $B$ are equivalent. In other words the categories $X_{app}$ and $Z_{app}$ of Lemma 61.12.21 are equivalent. We omit the verification that this equivalence sends coverings to coverings and vice versa. Thus the result as Lemma 61.12.21 tells us the pro-étale topos is the topos of sheaves on $X_{app}$. $\square$

Lemma 61.25.5. Let $i : Z \to X$ be a closed immersion of schemes. Let $U \to X$ be an object of $X_{pro\text{-}\acute{e}tale}$ such that

  1. $U$ is affine and weakly contractible, and

  2. every point of $U$ specializes to a point of $U \times _ X Z$.

Then $i_{pro\text{-}\acute{e}tale}^{-1}\mathcal{F}(U \times _ X Z) = \mathcal{F}(U)$ for all abelian sheaves on $X_{pro\text{-}\acute{e}tale}$.

Proof. Since pullback commutes with restriction, we may replace $X$ by $U$. Thus we may assume that $X$ is affine and weakly contractible and that every point of $X$ specializes to a point of $Z$. By Lemma 61.25.2 part (1) it suffices to show that $v(Z) = X$ in this case. Thus we have to show: If $A$ is a w-contractible ring, $I \subset A$ an ideal contained in the Jacobson radical of $A$ and $A \to B \to A/I$ is a factorization with $A \to B$ ind-étale, then there is a unique retraction $B \to A$ compatible with maps to $A/I$. Observe that $B/IB = A/I \times R$ as $A/I$-algebras. After replacing $B$ by a localization we may assume $B/IB = A/I$. Note that $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is surjective as the image contains $V(I)$ and hence all closed points and is closed under specialization. Since $A$ is w-contractible there is a retraction $B \to A$. Since $B/IB = A/I$ this retraction is compatible with the map to $A/I$. We omit the proof of uniqueness (hint: use that $A$ and $B$ have isomorphic local rings at maximal ideals of $A$). $\square$

Lemma 61.25.6. Let $i : Z \to X$ be a closed immersion of schemes. If $X \setminus i(Z)$ is a retrocompact open of $X$, then $i_{{pro\text{-}\acute{e}tale}, *}$ is exact.

Proof. The question is local on $X$ hence we may assume $X$ is affine. Say $X = \mathop{\mathrm{Spec}}(A)$ and $Z = \mathop{\mathrm{Spec}}(A/I)$. There exist $f_1, \ldots , f_ r \in I$ such that $Z = V(f_1, \ldots , f_ r)$ set theoretically, see Algebra, Lemma 10.29.1. By Lemma 61.25.4 we may assume that $Z = \mathop{\mathrm{Spec}}(A/(f_1, \ldots , f_ r))$. In this case the functor $i_{{pro\text{-}\acute{e}tale}, *}$ is exact by Lemma 61.24.1. $\square$

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 09A9. Beware of the difference between the letter 'O' and the digit '0'.