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18.19 Localization of ringed sites

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. We explain the counterparts of the results in Sites, Section 7.25 in this setting.

Denote $\mathcal{O}_ U = j_ U^{-1}\mathcal{O}$ the restriction of $\mathcal{O}$ to the site $\mathcal{C}/U$. It is described by the simple rule $\mathcal{O}_ U(V/U) = \mathcal{O}(V)$. With this notation the localization morphism $j_ U$ becomes a morphism of ringed topoi

\[ (j_ U, j_ U^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \]

namely, we take $j_ U^\sharp : j_ U^{-1}\mathcal{O} \to \mathcal{O}_ U$ the identity map. Moreover, we obtain the following descriptions for pushforward and pullback of modules.

Definition 18.19.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

  1. The ringed site $(\mathcal{C}/U, \mathcal{O}_ U)$ is called the localization of the ringed site $(\mathcal{C}, \mathcal{O})$ at the object $U$.

  2. The morphism of ringed topoi $(j_ U, j_ U^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ is called the localization morphism.

  3. The functor $j_{U*} : \textit{Mod}(\mathcal{O}_ U) \to \textit{Mod}(\mathcal{O})$ is called the direct image functor.

  4. For a sheaf of $\mathcal{O}$-modules $\mathcal{F}$ on $\mathcal{C}$ the sheaf $j_ U^*\mathcal{F}$ is called the restriction of $\mathcal{F}$ to $\mathcal{C}/U$. We will sometimes denote it by $\mathcal{F}|_{\mathcal{C}/U}$ or even $\mathcal{F}|_ U$. It is described by the simple rule $j_ U^*(\mathcal{F})(X/U) = \mathcal{F}(X)$.

  5. The left adjoint $j_{U!} : \textit{Mod}(\mathcal{O}_ U) \to \textit{Mod}(\mathcal{O})$ of restriction is called extension by zero. It exists and is exact by Lemmas 18.19.2 and 18.19.3.

As in the topological case, see Sheaves, Section 6.31, the extension by zero $j_{U!}$ functor is different from extension by the empty set $j_{U!}$ defined on sheaves of sets. Here is the lemma defining extension by zero.

Lemma 18.19.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. 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

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O}_ U)}(\mathcal{G}, j_ U^*\mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O})}(j_{U!}\mathcal{G}, \mathcal{F}) \]

for $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\textit{Mod}(\mathcal{O}))$ and $\mathcal{G} \in \mathop{\mathrm{Ob}}\nolimits (\textit{Mod}(\mathcal{O}_ U))$. Moreover, the extension by zero $j_{U!}\mathcal{G}$ of $\mathcal{G}$ is the sheaf associated to the presheaf

\[ V \longmapsto \bigoplus \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)} \mathcal{G}(V \xrightarrow {\varphi } U) \]

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

\[ \bigoplus \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)} \mathcal{G}(V \xrightarrow {\varphi } U) \longrightarrow \mathcal{F}(V) \]

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 7.25 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

\[ V \longmapsto \coprod \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)} \mathcal{G}(V \xrightarrow {\varphi } U) \]

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. $\square$

Note that we have in the situation of Definition 18.19.1 we have
\begin{equation} \label{sites-modules-equation-map-lower-shriek-OU-into-module} \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_{U!}\mathcal{O}_ U, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{O}_ U, j_ U^*\mathcal{F}) = \mathcal{F}(U) \end{equation}

for every $\mathcal{O}$-module $\mathcal{F}$. Namely, the first equality holds by the adjointness of $j_{U!}$ and $j_ U^*$ and the second because $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{O}_ U, j_ U^*\mathcal{F}) = j_ U^*\mathcal{F}(U/U) = \mathcal{F}|_ U(U/U) = \mathcal{F}(U)$.

Lemma 18.19.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. The functor $j_{U!} : \textit{Mod}(\mathcal{O}_ U) \to \textit{Mod}(\mathcal{O})$ is exact.

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

\[ \bigoplus \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)} \mathcal{G}_1(V \xrightarrow {\varphi } U) \longrightarrow \bigoplus \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)} \mathcal{G}_2(V \xrightarrow {\varphi } U) \]

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. $\square$

Lemma 18.19.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. A complex of $\mathcal{O}_ U$-modules $\mathcal{G}_1 \to \mathcal{G}_2 \to \mathcal{G}_3$ is exact if and only if $j_{U!}\mathcal{G}_1 \to j_{U!}\mathcal{G}_2 \to j_{U!}\mathcal{G}_3$ is exact as a sequence of $\mathcal{O}$-modules.

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

\[ \mathcal{G}(V/U) \longrightarrow \bigoplus \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)} \mathcal{G}(V \xrightarrow {\varphi } U) \]

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. $\square$

Lemma 18.19.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $f : V \to U$ be a morphism of $\mathcal{C}$. Then there exists a commutative diagram

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V), \mathcal{O}_ V) \ar[rd]_{(j_ V, j_ V^\sharp )} \ar[rr]_{(j, j^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \ar[ld]^{(j_ U, j_ U^\sharp )} \\ & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) & } \]

of ringed topoi. Here $(j, j^\sharp )$ is the localization morphism associated to the object $V/U$ of the ringed site $(\mathcal{C}/V, \mathcal{O}_ V)$.

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 ( We omit the verification of the equality. $\square$

Remark 18.19.6. In the situation of Lemma 18.19.2 the diagram

\[ \xymatrix{ \textit{Mod}(\mathcal{O}_ U) \ar[r]_{j_{U!}} \ar[d]_{forget} & \textit{Mod}(\mathcal{O}_\mathcal {C}) \ar[d]^{forget} \\ \textit{Ab}(\mathcal{C}/U) \ar[r]^{j^{Ab}_{U!}} & \textit{Ab}(\mathcal{C}) } \]

commutes. This is clear from the explicit description of the functor $j_{U!}$ in the lemma.

Remark 18.19.7. Localization and presheaves of modules; see Sites, Remark 7.25.10. Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings. Let $U$ be an object of $\mathcal{C}$. Strictly speaking the functors $j_ U^*$, $j_{U*}$ and $j_{U!}$ have not been defined for presheaves of $\mathcal{O}$-modules. But of course, we can think of a presheaf as a sheaf for the chaotic topology on $\mathcal{C}$ (see Sites, Examples 7.6.6). Hence we also obtain a functor

\[ j_ U^* : \textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O}_ U) \]

and functors

\[ j_{U*}, j_{U!} : \textit{PMod}(\mathcal{O}_ U) \longrightarrow \textit{PMod}(\mathcal{O}) \]

which are right, left adjoint to $j_ U^*$. Inspecting the proof of Lemma 18.19.2 we see that $j_{U!}\mathcal{G}$ is the presheaf

\[ V \longmapsto \bigoplus \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)} \mathcal{G}(V \xrightarrow {\varphi } U) \]

In addition the functor $j_{U!}$ is exact (by Lemma 18.19.3 in the case of the discrete topologies). Moreover, if $\mathcal{C}$ is actually a site, and $\mathcal{O}$ is actually a sheaf of rings, then the diagram

\[ \xymatrix{ \textit{Mod}(\mathcal{O}_ U) \ar[r]_{j_{U!}} \ar[d]_{forget} & \textit{Mod}(\mathcal{O}) \\ \textit{PMod}(\mathcal{O}_ U) \ar[r]^{j_{U!}} & \textit{PMod}(\mathcal{O}) \ar[u]_{(\ )^\# } } \]


Lemma 18.19.8. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Assume that every $X$ in $\mathcal{C}$ has at most one morphism to $U$. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}/U$. The canonical maps $\mathcal{F} \to j_ U^{-1}j_{U!}\mathcal{F}$ and $j_ U^{-1}j_{U*}\mathcal{F} \to \mathcal{F}$ are isomorphisms.

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 $\mathcal{C}/U \to \mathcal{C}$. $\square$

Comments (2)

Comment #2474 by on

In the statement of Lemma 18.19.4, I think that maybe it should be the object of the ringed site , since the map is .

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