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
$i^{-1}\mathcal{F}$ 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, 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.
$\square$
Lemma 61.25.3. Let $i : Z \to X$ be a closed immersion of schemes. Then
$i_{pro\text{-}\acute{e}tale}^{-1}$ commutes with limits,
$i_{{pro\text{-}\acute{e}tale}, *}$ is fully faithful, and
$i_{pro\text{-}\acute{e}tale}^{-1}i_{{pro\text{-}\acute{e}tale}, *} \cong \text{id}_{\mathop{\mathit{Sh}}\nolimits (Z_{pro\text{-}\acute{e}tale})}$.
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
$U$ is affine and weakly contractible, and
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.
$\square$
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