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Tag 03DI

Chapter 18: Modules on Sites > Section 18.19: Localization of ringed sites

Lemma 18.19.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$. The restriction functor $j_U^* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_U)$ has a left adjoint $j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$. So $$ \mathop{\rm Mor}\nolimits_{\textit{Mod}(\mathcal{O}_U)}(\mathcal{G}, j_U^*\mathcal{F}) = \mathop{\rm Mor}\nolimits_{\textit{Mod}(\mathcal{O})}(j_{U!}\mathcal{G}, \mathcal{F}) $$ for $\mathcal{F} \in \mathop{\rm Ob}\nolimits(\textit{Mod}(\mathcal{O}))$ and $\mathcal{G} \in \mathop{\rm Ob}\nolimits(\textit{Mod}(\mathcal{O}_U))$. Moreover, the extension by zero $j_{U!}\mathcal{G}$ of $\mathcal{G}$ is the sheaf associated to the presheaf $$ V \longmapsto \bigoplus\nolimits_{\varphi \in \mathop{\rm Mor}\nolimits_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U) $$ with obvious restriction mappings and an obvious $\mathcal{O}$-module structure.

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 $$ \bigoplus\nolimits_{\varphi \in \mathop{\rm Mor}\nolimits_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U) \longrightarrow \mathcal{F}(V) $$ 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 7.24 that there exists an extension by the empty set $j^{Sh}_{U!} : \mathop{\textit{Sh}}\nolimits(\mathcal{C}/U) \to \mathop{\textit{Sh}}\nolimits(\mathcal{C})$ 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 $$ V \longmapsto \coprod\nolimits_{\varphi \in \mathop{\rm Mor}\nolimits_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U) $$ 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. $\square$

    The code snippet corresponding to this tag is a part of the file sites-modules.tex and is located in lines 2049–2075 (see updates for more information).

    \begin{lemma}
    \label{lemma-extension-by-zero}
    Let $(\mathcal{C}, \mathcal{O})$ be a ringed site.
    Let $U \in \Ob(\mathcal{C})$.
    The restriction functor
    $j_U^* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_U)$
    has a left adjoint
    $j_{U!} : \textit{Mod}(\mathcal{O}_U) \to \textit{Mod}(\mathcal{O})$.
    So
    $$
    \Mor_{\textit{Mod}(\mathcal{O}_U)}(\mathcal{G}, j_U^*\mathcal{F})
    =
    \Mor_{\textit{Mod}(\mathcal{O})}(j_{U!}\mathcal{G}, \mathcal{F})
    $$
    for $\mathcal{F} \in \Ob(\textit{Mod}(\mathcal{O}))$
    and $\mathcal{G} \in \Ob(\textit{Mod}(\mathcal{O}_U))$.
    Moreover, the extension by zero $j_{U!}\mathcal{G}$ of $\mathcal{G}$
    is the sheaf associated to the presheaf
    $$
    V
    \longmapsto
    \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)}
    \mathcal{G}(V \xrightarrow{\varphi} U)
    $$
    with obvious restriction mappings and an obvious $\mathcal{O}$-module
    structure.
    \end{lemma}
    
    \begin{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
    $\mathcal{C}/U$).
    
    \medskip\noindent
    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
    $$
    \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)}
    \mathcal{G}(V \xrightarrow{\varphi} U)
    \longrightarrow
    \mathcal{F}(V)
    $$
    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 \ref{lemma-sheafification-presheaf-modules}).
    
    \medskip\noindent
    Conversely, let $\beta : j_{U!}\mathcal{G} \to \mathcal{F}$
    by a map of $\mathcal{O}$-modules.
    Recall from Sites, Section \ref{sites-section-localize}
    that there exists an extension by the empty set
    $j^{Sh}_{U!} : \Sh(\mathcal{C}/U) \to \Sh(\mathcal{C})$
    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
    $$
    V
    \longmapsto
    \coprod\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)}
    \mathcal{G}(V \xrightarrow{\varphi} U)
    $$
    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.
    
    \medskip\noindent
    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.
    \end{proof}

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