The Stacks project

18.40 Locally ringed topoi

A reference for this section is [Exposé IV, Exercice 13.9, SGA4].

Lemma 18.40.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. The following are equivalent

  1. For every object $U$ of $\mathcal{C}$ and $f \in \mathcal{O}(U)$ there exists a covering $\{ U_ j \to U\} $ such that for each $j$ either $f|_{U_ j}$ is invertible or $(1 - f)|_{U_ j}$ is invertible.

  2. For $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, $n \geq 1$, and $f_1, \ldots , f_ n \in \mathcal{O}(U)$ which generate the unit ideal in $\mathcal{O}(U)$ there exists a covering $\{ U_ j \to U\} $ such that for each $j$ there exists an $i$ such that $f_ i|_{U_ j}$ is invertible.

  3. The map of sheaves of sets

    \[ (\mathcal{O} \times \mathcal{O}) \amalg (\mathcal{O} \times \mathcal{O}) \longrightarrow \mathcal{O} \times \mathcal{O} \]

    which maps $(f, a)$ in the first component to $(f, af)$ and $(f, b)$ in the second component to $(f, b(1 - f))$ is surjective.

Proof. It is clear that (2) implies (1). To show that (1) implies (2) we argue by induction on $n$. The first case is $n = 2$ (since $n = 1$ is trivial). In this case we have $a_1f_1 + a_2f_2 = 1$ for some $a_1, a_2 \in \mathcal{O}(U)$. By assumption we can find a covering $\{ U_ j \to U\} $ such that for each $j$ either $a_1f_1|_{U_ j}$ is invertible or $a_2f_2|_{U_ j}$ is invertible. Hence either $f_1|_{U_ j}$ is invertible or $f_2|_{U_ j}$ is invertible as desired. For $n > 2$ we have $a_1f_1 + \ldots + a_ nf_ n = 1$ for some $a_1, \ldots , a_ n \in \mathcal{O}(U)$. By the case $n = 2$ we see that we have some covering $\{ U_ j \to U\} _{j \in J}$ such that for each $j$ either $f_ n|_{U_ j}$ is invertible or $a_1f_1 + \ldots + a_{n - 1}f_{n - 1}|_{U_ j}$ is invertible. Say the first case happens for $j \in J_ n$. Set $J' = J \setminus J_ n$. By induction hypothesis, for each $j \in J'$ we can find a covering $\{ U_{jk} \to U_ j\} _{k \in K_ j}$ such that for each $k \in K_ j$ there exists an $i \in \{ 1, \ldots , n - 1\} $ such that $f_ i|_{U_{jk}}$ is invertible. By the axioms of a site the family of morphisms $\{ U_ j \to U\} _{j \in J_ n} \cup \{ U_{jk} \to U\} _{j \in J', k \in K_ j}$ is a covering which has the desired property.

Assume (1). To see that the map in (3) is surjective, let $(f, c)$ be a section of $\mathcal{O} \times \mathcal{O}$ over $U$. By assumption there exists a covering $\{ U_ j \to U\} $ such that for each $j$ either $f$ or $1 - f$ restricts to an invertible section. In the first case we can take $a = c|_{U_ j} (f|_{U_ j})^{-1}$, and in the second case we can take $b = c|_{U_ j} (1 - f|_{U_ j})^{-1}$. Hence $(f, c)$ is in the image of the map on each of the members. Conversely, assume (3) holds. For any $U$ and $f \in \mathcal{O}(U)$ there exists a covering $\{ U_ j \to U\} $ of $U$ such that the section $(f, 1)|_{U_ j}$ is in the image of the map in (3) on sections over $U_ j$. This means precisely that either $f$ or $1 - f$ restricts to an invertible section over $U_ j$, and we see that (1) holds. $\square$

Lemma 18.40.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Consider the following conditions

  1. For every object $U$ of $\mathcal{C}$ and $f \in \mathcal{O}(U)$ there exists a covering $\{ U_ j \to U\} $ such that for each $j$ either $f|_{U_ j}$ is invertible or $(1 - f)|_{U_ j}$ is invertible.

  2. For every point $p$ of $\mathcal{C}$ the stalk $\mathcal{O}_ p$ is either the zero ring or a local ring.

We always have (1) $\Rightarrow $ (2). If $\mathcal{C}$ has enough points then (1) and (2) are equivalent.

Proof. Assume (1). Let $p$ be a point of $\mathcal{C}$ given by a functor $u : \mathcal{C} \to \textit{Sets}$. Let $f_ p \in \mathcal{O}_ p$. Since $\mathcal{O}_ p$ is computed by Sites, Equation (7.32.1.1) we may represent $f_ p$ by a triple $(U, x, f)$ where $x \in U(U)$ and $f \in \mathcal{O}(U)$. By assumption there exists a covering $\{ U_ i \to U\} $ such that for each $i$ either $f$ or $1 - f$ is invertible on $U_ i$. Because $u$ defines a point of the site we see that for some $i$ there exists an $x_ i \in u(U_ i)$ which maps to $x \in u(U)$. By the discussion surrounding Sites, Equation (7.32.1.1) we see that $(U, x, f)$ and $(U_ i, x_ i, f|_{U_ i})$ define the same element of $\mathcal{O}_ p$. Hence we conclude that either $f_ p$ or $1 - f_ p$ is invertible. Thus $\mathcal{O}_ p$ is a ring such that for every element $a$ either $a$ or $1 - a$ is invertible. This means that $\mathcal{O}_ p$ is either zero or a local ring, see Algebra, Lemma 10.18.3.

Assume (2) and assume that $\mathcal{C}$ has enough points. Consider the map of sheaves of sets

\[ \mathcal{O} \times \mathcal{O} \amalg \mathcal{O} \times \mathcal{O} \longrightarrow \mathcal{O} \times \mathcal{O} \]

of Lemma 18.40.1 part (3). For any local ring $R$ the corresponding map $(R \times R) \amalg (R \times R) \to R \times R$ is surjective, see for example Algebra, Lemma 10.18.3. Since each $\mathcal{O}_ p$ is a local ring or zero the map is surjective on stalks. Hence, by our assumption that $\mathcal{C}$ has enough points it is surjective and we win. $\square$

In Modules, Section 17.2 we pointed out how in a ringed space $(X, \mathcal{O}_ X)$ there can be an open subspace over which the structure sheaf is zero. To prevent this we can require the sections $1$ and $0$ to have different values in every stalk of the space $X$. In the setting of ringed topoi and ringed sites the condition is that

18.40.2.1
\begin{equation} \label{sites-modules-equation-one-is-never-zero} \emptyset ^\# \longrightarrow \text{Equalizer}(0, 1 : * \longrightarrow \mathcal{O}) \end{equation}

is an isomorphism of sheaves. Here $*$ is the singleton sheaf, resp. $\emptyset ^\# $ is the “empty sheaf”, i.e., the final, resp. initial object in the category of sheaves, see Sites, Example 7.10.2, resp. Section 7.42. In other words, the condition is that whenever $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ is not sheaf theoretically empty, then $1, 0 \in \mathcal{O}(U)$ are not equal. Let us state the obligatory lemma.

Lemma 18.40.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Consider the statements

  1. (18.40.2.1) is an isomorphism, and

  2. for every point $p$ of $\mathcal{C}$ the stalk $\mathcal{O}_ p$ is not the zero ring.

We always have (1) $\Rightarrow $ (2) and if $\mathcal{C}$ has enough points then (1) $\Leftrightarrow $ (2).

Proof. Omitted. $\square$

Lemmas 18.40.1, 18.40.2, and 18.40.3 motivate the following definition.

Definition 18.40.4. A ringed site $(\mathcal{C}, \mathcal{O})$ is said to be locally ringed site if (18.40.2.1) is an isomorphism, and the equivalent properties of Lemma 18.40.1 are satisfied.

In [Exposé IV, Exercice 13.9, SGA4] the condition that (18.40.2.1) be an isomorphism is missing leading to a slightly different notion of a locally ringed site and locally ringed topos. As we are motivated by the notion of a locally ringed space we decided to add this condition (see explanation above).

Lemma 18.40.5. Being a locally ringed site is an intrinsic property. More precisely,

  1. if $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is a morphism of topoi and $(\mathcal{C}, \mathcal{O})$ is a locally ringed site, then $(\mathcal{C}', f^{-1}\mathcal{O})$ is a locally ringed site, and

  2. if $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ is an equivalence of ringed topoi, then $(\mathcal{C}, \mathcal{O})$ is locally ringed if and only if $(\mathcal{C}', \mathcal{O}')$ is locally ringed.

Proof. It is clear that (2) follows from (1). To prove (1) note that as $f^{-1}$ is exact we have $f^{-1}* = *$, $f^{-1}\emptyset ^\# = \emptyset ^\# $, and $f^{-1}$ commutes with products, equalizers and transforms isomorphisms and surjections into isomorphisms and surjections. Thus $f^{-1}$ transforms the isomorphism (18.40.2.1) into its analogue for $f^{-1}\mathcal{O}$ and transforms the surjection of Lemma 18.40.1 part (3) into the corresponding surjection for $f^{-1}\mathcal{O}$. $\square$

In fact Lemma 18.40.5 part (2) is the analogue of Schemes, Lemma 26.2.2. It assures us that the following definition makes sense.

Definition 18.40.6. A ringed topos $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ is said to be locally ringed if the underlying ringed site $(\mathcal{C}, \mathcal{O})$ is locally ringed.

Here is an example of a consequence of being locally ringed.

Lemma 18.40.7. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. Any locally free $\mathcal{O}$-module of rank $1$ is invertible. If $(\mathcal{C}, \mathcal{O})$ is locally ringed, then the converse holds as well (but in general this is not the case).

Proof. Assume $\mathcal{L}$ is locally free of rank $1$ and consider the evaluation map

\[ \mathcal{L} \otimes _\mathcal {O} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{L}, \mathcal{O}) \longrightarrow \mathcal{O} \]

Given any object $U$ of $\mathcal{C}$ and restricting to the members of a covering trivializing $\mathcal{L}$, we see that this map is an isomorphism (details omitted). Hence $\mathcal{L}$ is invertible by Lemma 18.32.2.

Assume $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ is locally ringed. Let $U$ be an object of $\mathcal{C}$. In the proof of Lemma 18.32.2 we have seen that there exists a covering $\{ U_ i \to U\} $ such that $\mathcal{L}|_{\mathcal{C}/U_ i}$ is a direct summand of a finite free $\mathcal{O}_{U_ i}$-module. After replacing $U$ by $U_ i$, let $p : \mathcal{O}_ U^{\oplus r} \to \mathcal{O}_ U^{\oplus r}$ be a projector whose image is isomorphic to $\mathcal{L}|_{\mathcal{C}/U}$. Then $p$ corresponds to a matrix

\[ P = (p_{ij}) \in \text{Mat}(r \times r, \mathcal{O}(U)) \]

which is a projector: $P^2 = P$. Set $A = \mathcal{O}(U)$ so that $P \in \text{Mat}(r \times r, A)$. By Algebra, Lemma 10.78.2 the image of $P$ is a finite locally free module $M$ over $A$. Hence there are $f_1, \ldots , f_ t \in A$ generating the unit ideal, such that $M_{f_ i}$ is finite free. By Lemma 18.40.1 after replacing $U$ by the members of an open covering, we may assume that $M$ is free. This means that $\mathcal{L}|_ U$ is free (details omitted). Of course, since $\mathcal{L}$ is invertible, this is only possible if the rank of $\mathcal{L}|_ U$ is $1$ and the proof is complete. $\square$

Next, we want to work out what it means to have a morphism of locally ringed spaces. In order to do this we have the following lemma.

Lemma 18.40.8. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Consider the following conditions

  1. The diagram of sheaves

    \[ \xymatrix{ f^{-1}(\mathcal{O}^*_\mathcal {D}) \ar[r]_-{f^\sharp } \ar[d] & \mathcal{O}^*_\mathcal {C} \ar[d] \\ f^{-1}(\mathcal{O}_\mathcal {D}) \ar[r]^-{f^\sharp } & \mathcal{O}_\mathcal {C} } \]

    is cartesian.

  2. For any point $p$ of $\mathcal{C}$, setting $q = f \circ p$, the diagram

    \[ \xymatrix{ \mathcal{O}^*_{\mathcal{D}, q} \ar[r] \ar[d] & \mathcal{O}^*_{\mathcal{C}, p} \ar[d] \\ \mathcal{O}_{\mathcal{D}, q} \ar[r] & \mathcal{O}_{\mathcal{C}, p} } \]

    of sets is cartesian.

We always have (1) $\Rightarrow $ (2). If $\mathcal{C}$ has enough points then (1) and (2) are equivalent. If $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ and $(\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ are locally ringed topoi then (2) is equivalent to

  1. For any point $p$ of $\mathcal{C}$, setting $q = f \circ p$, the ring map $\mathcal{O}_{\mathcal{D}, q} \to \mathcal{O}_{\mathcal{C}, p}$ is a local ring map.

In fact, properties (2), or (3) for a conservative family of points implies (1).

Proof. This lemma proves itself, in other words, it follows by unwinding the definitions. $\square$

Definition 18.40.9. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Assume $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ and $(\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ are locally ringed topoi. We say that $(f, f^\sharp )$ is a morphism of locally ringed topoi if and only if the diagram of sheaves

\[ \xymatrix{ f^{-1}(\mathcal{O}^*_\mathcal {D}) \ar[r]_-{f^\sharp } \ar[d] & \mathcal{O}^*_\mathcal {C} \ar[d] \\ f^{-1}(\mathcal{O}_\mathcal {D}) \ar[r]^-{f^\sharp } & \mathcal{O}_\mathcal {C} } \]

(see Lemma 18.40.8) is cartesian. If $(f, f^\sharp )$ is a morphism of ringed sites, then we say that it is a morphism of locally ringed sites if the associated morphism of ringed topoi is a morphism of locally ringed topoi.

It is clear that an isomorphism of ringed topoi between locally ringed topoi is automatically an isomorphism of locally ringed topoi.

Lemma 18.40.10. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_1), \mathcal{O}_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_2), \mathcal{O}_2)$ and $(g, g^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_2), \mathcal{O}_2) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_3), \mathcal{O}_3)$ be morphisms of locally ringed topoi. Then the composition $(g, g^\sharp ) \circ (f, f^\sharp )$ (see Definition 18.7.1) is also a morphism of locally ringed topoi.

Proof. Omitted. $\square$

Lemma 18.40.11. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a morphism of topoi. If $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}$, then

\[ f^{-1}(\mathcal{O}^*) = (f^{-1}\mathcal{O})^*. \]

In particular, if $\mathcal{O}$ turns $\mathcal{C}$ into a locally ringed site, then setting $f^\sharp = \text{id}$ the morphism of ringed topoi

\[ (f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), f^{-1}\mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \mathcal{O}) \]

is a morphism of locally ringed topoi.

Proof. Note that the diagram

\[ \xymatrix{ \mathcal{O}^* \ar[rr] \ar[d]_{u \mapsto (u, u^{-1})} & & {*} \ar[d]^{1} \\ \mathcal{O} \times \mathcal{O} \ar[rr]^-{(a, b) \mapsto ab} & & \mathcal{O} } \]

is cartesian. Since $f^{-1}$ is exact we conclude that

\[ \xymatrix{ f^{-1}(\mathcal{O}^*) \ar[d]_{u \mapsto (u, u^{-1})} \ar[rr] & & {*} \ar[d]^{1} \\ f^{-1}\mathcal{O} \times f^{-1}\mathcal{O} \ar[rr]^-{(a, b) \mapsto ab} & & f^{-1}\mathcal{O} } \]

is cartesian which implies the first assertion. For the second, note that $(\mathcal{C}', f^{-1}\mathcal{O})$ is a locally ringed site by Lemma 18.40.5 so that the assertion makes sense. Now the first part implies that the morphism is a morphism of locally ringed topoi. $\square$

Lemma 18.40.12. Localization of locally ringed sites and topoi.

  1. Let $(\mathcal{C}, \mathcal{O})$ be a locally ringed site. Let $U$ be an object of $\mathcal{C}$. Then the localization $(\mathcal{C}/U, \mathcal{O}_ U)$ is a locally ringed site, and the localization morphism

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

    is a morphism of locally ringed topoi.

  2. Let $(\mathcal{C}, \mathcal{O})$ be a locally ringed site. Let $f : V \to U$ be a morphism of $\mathcal{C}$. Then the morphism

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

    of Lemma 18.19.5 is a morphism of locally ringed topoi.

  3. Let $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')$ be a morphism of locally ringed sites where $f$ is given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V$ be an object of $\mathcal{D}$ and let $U = u(V)$. Then the morphism

    \[ (f', (f')^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V), \mathcal{O}'_ V) \]

    of Lemma 18.20.1 is a morphism of locally ringed sites.

  4. Let $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')$ be a morphism of locally ringed sites where $f$ is given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$, $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and $c : U \to u(V)$. Then the morphism

    \[ (f_ c, (f_ c)^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V), \mathcal{O}'_ V) \]

    of Lemma 18.20.2 is a morphism of locally ringed topoi.

  5. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a locally ringed topos. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Then the localization $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F})$ is a locally ringed topos and the localization morphism

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

    is a morphism of locally ringed topoi.

  6. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a locally ringed topos. Let $s : \mathcal{G} \to \mathcal{F}$ be a map of sheaves on $\mathcal{C}$. Then the morphism

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

    of Lemma 18.21.4 is a morphism of locally ringed topoi.

  7. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a morphism of locally ringed topoi. Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$. Set $\mathcal{F} = f^{-1}\mathcal{G}$. Then the morphism

    \[ (f', (f')^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G}, \mathcal{O}'_\mathcal {G}) \]

    of Lemma 18.22.1 is a morphism of locally ringed topoi.

  8. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a morphism of locally ringed topoi. Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$, let $\mathcal{F}$ be a sheaf on $\mathcal{C}$, and let $s : \mathcal{F} \to f^{-1}\mathcal{G}$ be a morphism of sheaves. Then the morphism

    \[ (f_ s, (f_ s)^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G}, \mathcal{O}'_\mathcal {G}) \]

    of Lemma 18.22.3 is a morphism of locally ringed topoi.

Proof. Part (1) is clear since $\mathcal{O}_ U$ is just the restriction of $\mathcal{O}$, so Lemmas 18.40.5 and 18.40.11 apply. Part (2) is clear as the morphism $(j, j^\sharp )$ is actually a localization of a locally ringed site so (1) applies. Part (3) is clear also since $(f')^\sharp $ is just the restriction of $f^\sharp $ to the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$, see proof of Lemma 18.22.1 (hence the diagram of Definition 18.40.9 for the morphism $f'$ is just the restriction of the corresponding diagram for $f$, and restriction is an exact functor). Part (4) follows formally on combining (2) and (3). Parts (5), (6), (7), and (8) follow from their counterparts (1), (2), (3), and (4) by enlarging the sites as in Lemma 18.7.2 and translating everything in terms of sites and morphisms of sites using the comparisons of Lemmas 18.21.3, 18.21.5, 18.22.2, and 18.22.4. (Alternatively one could use the same arguments as in the proofs of (1), (2), (3), and (4) to prove (5), (6), (7), and (8) directly.) $\square$

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