The Stacks project

Proof. The $\mathcal{O}$-module structure on the presheaf is defined as follows. If $f \in \mathcal{O}(V)$ and $s \in \mathcal{G}(V \xrightarrow {\varphi } U)$, then we define $f \cdot s = fs$ where $f \in \mathcal{O}_ U(\varphi : V \to U) = \mathcal{O}(V)$ (because $\mathcal{O}_ U$ is the restriction of $\mathcal{O}$ to $\mathcal{C}/U$).

Similarly, let $\alpha : \mathcal{G} \to \mathcal{F}|_ U$ be a morphism of $\mathcal{O}_ U$-modules. In this case we can define a map from the presheaf of the lemma into $\mathcal{F}$ by mapping

by the rule that $s \in \mathcal{G}(V \xrightarrow {\varphi } U)$ maps to $\alpha (s) \in \mathcal{F}(V)$. It is clear that this is $\mathcal{O}$-linear, and hence induces a morphism of $\mathcal{O}$-modules $\alpha ' : j_{U!}\mathcal{G} \to \mathcal{F}$ by the properties of sheafification of modules (Lemma 18.11.1).

Conversely, let $\beta : j_{U!}\mathcal{G} \to \mathcal{F}$ by a map of $\mathcal{O}$-modules. Recall from Sites, Section 7.25 that there exists an extension by the empty set $j^{Sh}_{U!} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ on sheaves of sets which is left adjoint to $j_ U^{-1}$. Moreover, $j^{Sh}_{U!}\mathcal{G}$ is the sheaf associated to the presheaf

Hence there is a natural map $j^{Sh}_{U!}\mathcal{G} \to j_{U!}\mathcal{G}$ of sheaves of sets. Hence precomposing $\beta$ by this map we get a map of sheaves of sets $j^{Sh}_{U!}\mathcal{G} \to \mathcal{F}$ which by adjunction corresponds to a map of sheaves of sets $\beta ' : \mathcal{G} \to \mathcal{F}|_ U$. We claim that $\beta '$ is $\mathcal{O}_ U$-linear. Namely, suppose that $\varphi : V \to U$ is an object of $\mathcal{C}/U$ and that $s, s' \in \mathcal{G}(\varphi : V \to U)$, and $f \in \mathcal{O}(V) = \mathcal{O}_ U(\varphi : V \to U)$. Then by the discussion above we see that $\beta '(s + s')$, resp. $\beta '(fs)$ in $\mathcal{F}|_ U(\varphi : V \to U)$ correspond to $\beta (s + s')$, resp. $\beta (fs)$ in $\mathcal{F}(V)$. Since $\beta$ is a homomorphism we conclude.

To conclude the proof of the lemma we have to show that the constructions $\alpha \mapsto \alpha '$ and $\beta \mapsto \beta '$ are mutually inverse. We omit the verifications. $\square$

Proof. Since $j_{U!}$ is a left adjoint to $j_ U^*$ we see that it is right exact (see Categories, Lemma 4.24.6 and Homology, Section 12.7). Hence it suffices to show that if $\mathcal{G}_1 \to \mathcal{G}_2$ is an injective map of $\mathcal{O}_ U$-modules, then $j_{U!}\mathcal{G}_1 \to j_{U!}\mathcal{G}_2$ is injective. The map on sections of presheaves over an object $V$ (as in Lemma 18.19.2) is the map

which is injective by assumption. Since sheafification is exact by Lemma 18.11.2 we conclude $j_{U!}\mathcal{G}_1 \to j_{U!}\mathcal{G}_2$ is injective and we win. $\square$

Proof. We already know that $j_{U!}$ is exact, see Lemma 18.19.3. Thus it suffices to show that $j_{U!} : \textit{Mod}(\mathcal{O}_ U) \to \textit{Mod}(\mathcal{O})$ reflects injections and surjections.

For every $\mathcal{G}$ in $\textit{Mod}(\mathcal{O}_ U)$ we have the unit $\mathcal{G} \to j_ U^*j_{U!}\mathcal{G}$ of the adjunction. We claim this map is an injection of sheaves. Namely, looking at the construction of Lemma 18.19.2 we see that this map is the sheafification of the rule sending the object $V/U$ of $\mathcal{C}/U$ to the injective map

given by the inclusion of the summand corresponding to the structure morphism $V \to U$. Since sheafification is exact the claim follows. Some details omitted.

If $\mathcal{G} \to \mathcal{G}'$ is a map of $\mathcal{O}_ U$-modules with $j_{U!}\mathcal{G} \to j_{U!}\mathcal{G}'$ injective, then $j_ U^*j_{U!}\mathcal{G} \to j_ U^*j_{U!}\mathcal{G}'$ is injective (restriction is exact), hence $\mathcal{G} \to j_ U^*j_{U!}\mathcal{G}'$ is injective, hence $\mathcal{G} \to \mathcal{G}'$ is injective. We conclude that $j_{U!}$ reflects injections.

Let $a : \mathcal{G} \to \mathcal{G}'$ be a map of $\mathcal{O}_ U$-modules such that $j_{U!}\mathcal{G} \to j_{U!}\mathcal{G}'$ is surjective. Let $\mathcal{H}$ be the cokernel of $a$. Then $j_{U!}\mathcal{H} = 0$ as $j_{U!}$ is exact. By the above the map $\mathcal{H} \to j^*_ U j_{U!}\mathcal{H}$ is injective. Hence $\mathcal{H} = 0$ as desired. $\square$

Proof. The only thing to check is that $j_ V^\sharp = j^\sharp \circ j^{-1}(j_ U^\sharp )$, since everything else follows directly from Sites, Lemma 7.25.8 and Sites, Equation (7.25.8.1). We omit the verification of the equality. $\square$

Proof. This is a special case of Lemma 18.16.4 because the assumption on $U$ is equivalent to the fully faithfulness of the localization functor $\mathcal{C}/U \to \mathcal{C}$. $\square$