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54.69 Extension by zero

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

Definition 54.69.1. Let $j : U \to X$ be an étale morphism of schemes.

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

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

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

If $\mathcal{F}$ is an abelian sheaf on $X_{\acute{e}tale}$, then $j_!\mathcal{F} \not= j_!^{Sh}\mathcal{F}$ in general. On the other hand $j_!$ for sheaves of $\Lambda $-modules agrees with $j_!$ on underlying abelian sheaves (Modules on Sites, Remark 18.19.6). The functor $j_!$ is characterized by the functorial isomorphism

\[ \mathop{\mathrm{Hom}}\nolimits _ X(j_!\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits _ U(\mathcal{F}, j^{-1}\mathcal{G}) \]

for all $\mathcal{F} \in \textit{Ab}(U_{\acute{e}tale})$ and $\mathcal{G} \in \textit{Ab}(X_{\acute{e}tale})$. Similarly for sheaves of $\Lambda $-modules.

To describe it more explicitly, recall that $j^{-1}$ is just the restriction via the functor $U_{\acute{e}tale}\to X_{\acute{e}tale}$. In other words, $j^{-1}\mathcal{G}(U') = \mathcal{G}(U')$ for $U'$ étale over $U$. For $\mathcal{F} \in \textit{Ab}(U_{\acute{e}tale})$ we consider the presheaf

\[ j_!^{PSh}\mathcal{F} : X_{\acute{e}tale}\longrightarrow \textit{Ab}, \quad V \longmapsto \bigoplus \nolimits _{V \to U} \mathcal{F}(V) \]

Then $j_!\mathcal{F}$ is the sheafification of $j_!^{PSh}\mathcal{F}$.

Exercise 54.69.2. Prove directly that $j_!$ is left adjoint to $j^{-1}$ and that $j_*$ is right adjoint to $j^{-1}$.

Proposition 54.69.3. Let $j : U \to X$ be an étale morphism of schemes. Let $\mathcal{F}$ in $\textit{Ab}(U_{\acute{e}tale})$. If $\overline{x} : \mathop{\mathrm{Spec}}(k) \to X$ is a geometric point of $X$, then

\[ (j_!\mathcal{F})_{\overline{x}} = \bigoplus \nolimits _{\overline{u} : \mathop{\mathrm{Spec}}(k) \to U,\ j(\overline{u}) = \overline{x}} \mathcal{F}_{\bar{u}}. \]

In particular, $j_!$ is an exact functor.

Proof. Exactness of $j_!$ is very general, see Modules on Sites, Lemma 18.19.3. Of course it does also follow from the description of stalks. The formula for the stalk of $j_!\mathcal{F}$ can be deduced directly from the explicit description of $j_!$ given above. On the other hand, we can deduce it from the very general Modules on Sites, Lemma 18.37.1 and the description of points of the small étale site in terms of geometric points, see Lemma 54.29.12. $\square$

Lemma 54.69.4 (Extension by zero commutes with base change). Let $f: Y \to X$ be a morphism of schemes. Let $j: V \to X$ be an étale morphism. Consider the fibre product

\[ \xymatrix{ V' = Y \times _ X V \ar[d]_{f'} \ar[r]_-{j'} & Y \ar[d]^ f \\ V \ar[r]^ j & X } \]

Then we have $j'_! f'^{-1} = f^{-1} j_!$ on abelian sheaves and on sheaves of modules.

Proof. This is true because $j'_! f'^{-1}$ is left adjoint to $f'_* (j')^{-1}$ and $f^{-1} j_!$ is left adjoint to $j^{-1}f_*$. Further $f'_* (j')^{-1} = j^{-1}f_*$ because $f_*$ commutes with étale localization (by construction). In fact, the lemma holds very generally in the setting of a morphism of sites, see Modules on Sites, Lemma 18.20.1. $\square$

Lemma 54.69.5. Let $j : U \to X$ be finite and étale. Then $j_! = j_*$ on abelian sheaves and sheaves of $\Lambda $-modules.

Proof. We prove this in the case of abelian sheaves. We claim there is a natural transformation $j_! \to j_*$. We will construct a canonical map

\[ j_!^{PSh}\mathcal{F} \to j_*\mathcal{F} \]

of functors $X_{\acute{e}tale}\to \textit{Ab}$ for any abelian sheaf $\mathcal{F}$ on $U_{\acute{e}tale}$. Sheafification of this map will be the desired map $j_!\mathcal{F} \to j_*\mathcal{F}$. Namely, given $V \to X$ étale we have

\[ j_!^{PSh}\mathcal{F}(V) = \bigoplus \nolimits _{\varphi : V \to U} \mathcal{F}(V \xrightarrow {\varphi } U) \quad \text{and}\quad j_*\mathcal{F}(V) = \mathcal{F}(V \times _ X U) \]

For each $\varphi $ we have an open and closed immersion

\[ \Gamma _\varphi = (1, \varphi ) : V \longrightarrow V \times _ X U \]

over $U$. (It is open as it is a morphism between schemes étale over $U$ and it is closed as it is a section of a scheme finite over $V$.) Thus for a section $s_\varphi \in \mathcal{F}(V \xrightarrow {\varphi } U)$ there exists a unique section $s'_\varphi $ in $\mathcal{F}(V \times _ X U)$ which pulls back to $s_\varphi $ by $\Gamma _\varphi $ and which restricts to zero on the complement of the image of $\Gamma _\varphi $. Then we map $(s_\varphi )$ in $j_!^{PSh}\mathcal{F}(V)$ to $\sum _\varphi s'_\varphi $ in $j_*\mathcal{F}(V) = \mathcal{F}(V \times _ X U)$. We leave it to the reader to see that this construction is compatible with restriction mappings.

It suffices to check $j_!\mathcal{F} \to j_*\mathcal{F}$ is an isomorphism étale locally on $X$. Thus we may assume $U \to X$ is a finite disjoint union of isomorphisms, see Étale Morphisms, Lemma 40.18.3. We omit the proof in this case. $\square$

Lemma 54.69.6. 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. For every abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ 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 $X_{\acute{e}tale}$.

Proof. We obtain the maps by the adjointness properties of the functors involved. For a geometric point $\overline{x}$ in $X$ we have either $\overline{x} \in U$ in which case the map on the left hand side is an isomorphism on stalks and the stalk of $i_*i^{-1}\mathcal{F}$ is zero or $\overline{x} \in Z$ in which case the map on the right hand side is an isomorphism on stalks and the stalk of $j_!j^{-1}\mathcal{F}$ is zero. Here we have used the description of stalks of Lemma 54.46.3 and Proposition 54.69.3. $\square$

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