18.25 Closed immersions of ringed topoi
When do we declare a morphism of ringed topoi i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') to be a closed immersion? By analogy with the discussion in Modules, Section 17.13 it seems natural to assume at least:
The functor i is a closed immersion of topoi (Sites, Definition 7.43.7).
The associated map \mathcal{O}' \to i_*\mathcal{O} is surjective.
These conditions already imply a number of pleasing results which we discuss in this section. However, it seems prudent to not actually define the notion of a closed immersion of ringed topoi as there are many different definitions we could use.
Lemma 18.25.1. Let i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') be a morphism of ringed topoi. Assume i is a closed immersion of topoi and i^\sharp : \mathcal{O}' \to i_*\mathcal{O} is surjective. Denote \mathcal{I} \subset \mathcal{O}' the kernel of i^\sharp . The functor
i_* : \textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}')
is exact, fully faithful, with essential image those \mathcal{O}'-modules \mathcal{G} such that \mathcal{I}\mathcal{G} = 0.
Proof.
By Lemma 18.15.2 and Sites, Lemma 7.43.8 we see that i_* is exact. From the fact that i_* is fully faithful on sheaves of sets, and the fact that i^\sharp is surjective it follows that i_* is fully faithful as a functor \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}'). Namely, suppose that \alpha : i_*\mathcal{F}_1 \to i_*\mathcal{F}_2 is an \mathcal{O}'-module map. By the fully faithfulness of i_* we obtain a map \beta : \mathcal{F}_1 \to \mathcal{F}_2 of sheaves of sets. To prove \beta is a map of modules we have to show that
\xymatrix{ \mathcal{O} \times \mathcal{F}_1 \ar[r] \ar[d] & \mathcal{F}_1 \ar[d] \\ \mathcal{O} \times \mathcal{F}_2 \ar[r] & \mathcal{F}_2 }
commutes. It suffices to prove commutativity after applying i_*. Consider
\xymatrix{ \mathcal{O}' \times i_*\mathcal{F}_1 \ar[r] \ar[d] & i_*\mathcal{O} \times i_*\mathcal{F}_1 \ar[r] \ar[d] & i_*\mathcal{F}_1 \ar[d] \\ \mathcal{O}' \times i_*\mathcal{F}_2 \ar[r] & i_*\mathcal{O} \times i_*\mathcal{F}_2 \ar[r] & i_*\mathcal{F}_2 }
We know the outer rectangle commutes. Since i^\sharp is surjective we conclude.
To finish the proof we have to prove the statement on the essential image of i_*. It is clear that i_*\mathcal{F} is annihilated by \mathcal{I} for any \mathcal{O}-module \mathcal{F}. Conversely, let \mathcal{G} be a \mathcal{O}'-module with \mathcal{I}\mathcal{G} = 0. By definition of a closed subtopos there exists a subsheaf \mathcal{U} of the final object of \mathcal{D} such that the essential image of i_* on sheaves of sets is the class of sheaves of sets \mathcal{H} such that \mathcal{H} \times \mathcal{U} \to \mathcal{U} is an isomorphism. In particular, i_*\mathcal{O} \times \mathcal{U} = \mathcal{U}. This implies that \mathcal{I} \times \mathcal{U} = \mathcal{O} \times \mathcal{U}. Hence our module \mathcal{G} satisfies \mathcal{G} \times \mathcal{U} = \{ 0\} \times \mathcal{U} = \mathcal{U} (because the zero module is isomorphic to the final object of sheaves of sets). Thus there exists a sheaf of sets \mathcal{F} on \mathcal{C} with i_*\mathcal{F} = \mathcal{G}. Since i_* is fully faithful on sheaves of sets, we see that in order to define the addition \mathcal{F} \times \mathcal{F} \to \mathcal{F} and the multiplication \mathcal{O} \times \mathcal{F} \to \mathcal{F} it suffices to use the addition
\mathcal{G} \times \mathcal{G} \longrightarrow \mathcal{G}
(given to us as \mathcal{G} is a \mathcal{O}'-module) and the multiplication
i_*\mathcal{O} \times \mathcal{G} \to \mathcal{G}
which is given to us as we have the multiplication by \mathcal{O}' which annihilates \mathcal{I} by assumption and i_*\mathcal{O} = \mathcal{O}'/\mathcal{I}. By construction \mathcal{G} is isomorphic to the pushforward of the \mathcal{O}-module \mathcal{F} so constructed.
\square
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