The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

18.20 Localization of morphisms of ringed sites

This section is the analogue of Sites, Section 7.28.

Lemma 18.20.1. Let $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')$ be a morphism of 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 set $U = u(V)$. Then there is a canonical map of sheaves of rings $(f')^\sharp $ such that the diagram of Sites, Lemma 7.28.1 is turned into a commutative diagram of ringed topoi

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \ar[rr]_{(j_ U, j_ U^\sharp )} \ar[d]_{(f', (f')^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp )} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V), \mathcal{O}'_ V) \ar[rr]^{(j_ V, j_ V^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}'). } \]

Moreover, in this situation we have $f'_*j_ U^{-1} = j_ V^{-1}f_*$ and $f'_*j_ U^* = j_ V^*f_*$.

Proof. Just take $(f')^\sharp $ to be

\[ (f')^{-1}\mathcal{O}'_ V = (f')^{-1}j_ V^{-1}\mathcal{O}' = j_ U^{-1}f^{-1}\mathcal{O}' \xrightarrow {j_ U^{-1}f^\sharp } j_ U^{-1}\mathcal{O} = \mathcal{O}_ U \]

and everything else follows from Sites, Lemma 7.28.1. (Note that $j^{-1} = j^*$ on sheaves of modules if $j$ is a localization morphism, hence the first equality of functors implies the second.) $\square$

Lemma 18.20.2. Let $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')$ be a morphism of 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 $c : U \to u(V)$ a morphism of $\mathcal{C}$. There exists a commutative diagram of ringed topoi

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \ar[rr]_{(j_ U, j_ U^\sharp )} \ar[d]_{(f_ c, f_ c^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp )} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V), \mathcal{O}'_ V) \ar[rr]^{(j_ V, j_ V^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}'). } \]

The morphism $(f_ c, f_ c^\sharp )$ is equal to the composition of the morphism

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

of Lemma 18.20.1 and the morphism

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

of Lemma 18.19.5. Given any morphisms $b : V' \to V$, $a : U' \to U$ and $c' : U' \to u(V')$ such that

\[ \xymatrix{ U' \ar[r]_-{c'} \ar[d]_ a & u(V') \ar[d]^{u(b)} \\ U \ar[r]^-c & u(V) } \]

commutes, then the following diagram of ringed topoi

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U'), \mathcal{O}_{U'}) \ar[rr]_{(j_{U'/U}, j_{U'/U}^\sharp )} \ar[d]_{(f_{c'}, f_{c'}^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \ar[d]^{(f_ c, f_ c^\sharp )} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V'), \mathcal{O}'_{V'}) \ar[rr]^{(j_{V'/V}, j_{V'/V}^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V), \mathcal{O}'_{V'}) } \]

commutes.

Proof. On the level of morphisms of topoi this is Sites, Lemma 7.28.3. To check that the diagrams commute as morphisms of ringed topoi use Lemmas 18.19.5 and 18.20.1 exactly as in the proof of Sites, Lemma 7.28.3. $\square$


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