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61.26 Extension by zero

The general material in Modules on Sites, Section 18.19 allows us to make the following definition.

Definition 61.26.1. Let $j : U \to X$ be a weakly étale morphism of schemes.

  1. The restriction functor $j^{-1} : \mathop{\mathit{Sh}}\nolimits (X_{pro\text{-}\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (U_{pro\text{-}\acute{e}tale})$ has a left adjoint $j_!^{Sh} : \mathop{\mathit{Sh}}\nolimits (X_{pro\text{-}\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (U_{pro\text{-}\acute{e}tale})$.

  2. The restriction functor $j^{-1} : \textit{Ab}(X_{pro\text{-}\acute{e}tale}) \to \textit{Ab}(U_{pro\text{-}\acute{e}tale})$ has a left adjoint which is denoted $j_! : \textit{Ab}(U_{pro\text{-}\acute{e}tale}) \to \textit{Ab}(X_{pro\text{-}\acute{e}tale})$ and called extension by zero.

  3. Let $\Lambda $ be a ring. The functor $j^{-1} : \textit{Mod}(X_{pro\text{-}\acute{e}tale}, \Lambda ) \to \textit{Mod}(U_{pro\text{-}\acute{e}tale}, \Lambda )$ has a left adjoint $j_! : \textit{Mod}(U_{pro\text{-}\acute{e}tale}, \Lambda ) \to \textit{Mod}(X_{pro\text{-}\acute{e}tale}, \Lambda )$ and called extension by zero.

As usual we compare this to what happens in the étale case.

Lemma 61.26.2. Let $j : U \to X$ be an étale morphism of schemes. Let $\mathcal{G}$ be an abelian sheaf on $U_{\acute{e}tale}$. Then $\epsilon ^{-1} j_!\mathcal{G} = j_!\epsilon ^{-1}\mathcal{G}$ as sheaves on $X_{pro\text{-}\acute{e}tale}$.

Proof. This is true because both are left adjoints to $j_{{pro\text{-}\acute{e}tale}, *}\epsilon ^{-1} = \epsilon ^{-1}j_{{\acute{e}tale}, *}$, see Lemma 61.23.1. $\square$

Lemma 61.26.3. Let $j : U \to X$ be a weakly étale morphism of schemes. Let $i : Z \to X$ be a closed immersion such that $U \times _ X Z = \emptyset $. Let $V \to X$ be an affine object of $X_{pro\text{-}\acute{e}tale}$ such that every point of $V$ specializes to a point of $V_ Z = Z \times _ X V$. Then $j_!\mathcal{F}(V) = 0$ for all abelian sheaves on $U_{pro\text{-}\acute{e}tale}$.

Proof. Let $\{ V_ i \to V\} $ be a pro-étale covering. The lemma follows if we can refine this covering to a covering where the members have no morphisms into $U$ over $X$ (see construction of $j_!$ in Modules on Sites, Section 18.19). First refine the covering to get a finite covering with $V_ i$ affine. For each $i$ let $V_ i = \mathop{\mathrm{Spec}}(A_ i)$ and let $Z_ i \subset V_ i$ be the inverse image of $Z$. Set $W_ i = \mathop{\mathrm{Spec}}(A_{i, Z_ i}^\sim )$ with notation as in Lemma 61.5.1. Then $\coprod W_ i \to V$ is weakly étale and the image contains all points of $V_ Z$. Hence the image contains all points of $V$ by our assumption on specializations. Thus $\{ W_ i \to V\} $ is a pro-étale covering refining the given one. But each point in $W_ i$ specializes to a point lying over $Z$, hence there are no morphisms $W_ i \to U$ over $X$. $\square$

Lemma 61.26.4. Let $j : U \to X$ be an open immersion of schemes. Then $\text{id} \cong j^{-1}j_!$ and $j^{-1}j_* \cong \text{id}$ and the functors $j_!$ and $j_*$ are fully faithful.

Proof. See Modules on Sites, Lemma 18.19.8 (and Sites, Lemma 7.27.4 for the case of sheaves of sets) and Categories, Lemma 4.24.4. $\square$

Here is the relationship between extension by zero and restriction to the complementary closed subscheme.

Lemma 61.26.5. Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme and let $U \subset X$ be the complement. Denote $i : Z \to X$ and $j : U \to X$ the inclusion morphisms. Assume that $j$ is a quasi-compact morphism. For every abelian sheaf on $X_{pro\text{-}\acute{e}tale}$ there is a canonical short exact sequence

\[ 0 \to j_!j^{-1}\mathcal{F} \to \mathcal{F} \to i_*i^{-1}\mathcal{F} \to 0 \]

on $X_{pro\text{-}\acute{e}tale}$ where all the functors are for the pro-étale topology.

Proof. We obtain the maps by the adjointness properties of the functors involved. It suffices to show that $X_{pro\text{-}\acute{e}tale}$ has enough objects (Sites, Definition 7.40.2) on which the sequence evaluates to a short exact sequence. Let $V = \mathop{\mathrm{Spec}}(A)$ be an affine object of $X_{pro\text{-}\acute{e}tale}$ such that $A$ is w-contractible (there are enough objects of this type). Then $V \times _ X Z$ is cut out by an ideal $I \subset A$. The assumption that $j$ is quasi-compact implies there exist $f_1, \ldots , f_ r \in I$ such that $V(I) = V(f_1, \ldots , f_ r)$. We obtain a faithfully flat, ind-Zariski ring map

\[ A \longrightarrow A_{f_1} \times \ldots \times A_{f_ r} \times A_{V(I)}^\sim \]

with $A_{V(I)}^\sim $ as in Lemma 61.5.1. Since $V_ i = \mathop{\mathrm{Spec}}(A_{f_ i}) \to X$ factors through $U$ we have

\[ j_!j^{-1}\mathcal{F}(V_ i) = \mathcal{F}(V_ i) \quad \text{and}\quad i_*i^{-1}\mathcal{F}(V_ i) = 0 \]

On the other hand, for the scheme $V^\sim = \mathop{\mathrm{Spec}}(A_{V(I)}^\sim )$ we have

\[ j_!j^{-1}\mathcal{F}(V^\sim ) = 0 \quad \text{and}\quad \mathcal{F}(V^\sim ) = i_*i^{-1}\mathcal{F}(V^\sim ) \]

the first equality by Lemma 61.26.3 and the second by Lemmas 61.25.5 and 61.11.7. Thus the sequence evaluates to an exact sequence on $\mathop{\mathrm{Spec}}(A_{f_1} \times \ldots \times A_{f_ r} \times A_{V(I)}^\sim )$ and the lemma is proved. $\square$

Lemma 61.26.6. Let $j : U \to X$ be a quasi-compact open immersion morphism of schemes. The functor $j_! : \textit{Ab}(U_{pro\text{-}\acute{e}tale}) \to \textit{Ab}(X_{pro\text{-}\acute{e}tale})$ commutes with limits.

Proof. Since $j_!$ is exact it suffices to show that $j_!$ commutes with products. The question is local on $X$, hence we may assume $X$ affine. Let $\mathcal{G}$ be an abelian sheaf on $U_{pro\text{-}\acute{e}tale}$. We have $j^{-1}j_*\mathcal{G} = \mathcal{G}$. Hence applying the exact sequence of Lemma 61.26.5 we get

\[ 0 \to j_!\mathcal{G} \to j_*\mathcal{G} \to i_*i^{-1}j_*\mathcal{G} \to 0 \]

where $i : Z \to X$ is the inclusion of the reduced induced scheme structure on the complement $Z = X \setminus U$. The functors $j_*$ and $i_*$ commute with products as right adjoints. The functor $i^{-1}$ commutes with products by Lemma 61.25.3. Hence $j_!$ does because on the pro-étale site products are exact (Cohomology on Sites, Proposition 21.49.2). $\square$

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