91.9 Thickenings of ringed topoi
This section is the analogue of Section 91.3 for ringed topoi. In the following few sections we will use the following notions:
A sheaf of ideals $\mathcal{I} \subset \mathcal{O}'$ on a ringed topos $(\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ is locally nilpotent if any local section of $\mathcal{I}$ is locally nilpotent.
A thickening of ringed topoi is a morphism $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ of ringed topoi such that
$i_*$ is an equivalence $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$,
the map $i^\sharp : \mathcal{O}' \to i_*\mathcal{O}$ is surjective, and
the kernel of $i^\sharp $ is a locally nilpotent sheaf of ideals.
A first order thickening of ringed topoi is a thickening $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ of ringed topoi such that $\mathop{\mathrm{Ker}}(i^\sharp )$ has square zero.
It is clear how to define morphisms of thickenings of ringed topoi, morphisms of thickenings of ringed topoi over a base ringed topos, etc.
If $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ is a thickening of ringed topoi then we identify the underlying topoi and think of $\mathcal{O}$, $\mathcal{O}'$, and $\mathcal{I} = \mathop{\mathrm{Ker}}(i^\sharp )$ as sheaves on $\mathcal{C}$. We obtain a short exact sequence
\[ 0 \to \mathcal{I} \to \mathcal{O}' \to \mathcal{O} \to 0 \]
of $\mathcal{O}'$-modules. By Modules on Sites, Lemma 18.25.1 the category of $\mathcal{O}$-modules is equivalent to the category of $\mathcal{O}'$-modules annihilated by $\mathcal{I}$. In particular, if $i$ is a first order thickening, then $\mathcal{I}$ is a $\mathcal{O}$-module.
Situation 91.9.1. A morphism of thickenings of ringed topoi $(f, f')$ is given by a commutative diagram
91.9.1.1
\begin{equation} \label{defos-equation-morphism-thickenings-ringed-topoi} \vcenter { \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \ar[r]_ i \ar[d]_ f & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') \ar[d]^{f'} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}), \mathcal{O}_\mathcal {B}) \ar[r]^ t & (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}'), \mathcal{O}_{\mathcal{B}'}) } } \end{equation}
of ringed topoi whose horizontal arrows are thickenings. In this situation we set $\mathcal{I} = \mathop{\mathrm{Ker}}(i^\sharp ) \subset \mathcal{O}'$ and $\mathcal{J} = \mathop{\mathrm{Ker}}(t^\sharp ) \subset \mathcal{O}_{\mathcal{B}'}$. As $f = f'$ on underlying topoi we will identify the pullback functors $f^{-1}$ and $(f')^{-1}$. Observe that $(f')^\sharp : f^{-1}\mathcal{O}_{\mathcal{B}'} \to \mathcal{O}'$ induces in particular a map $f^{-1}\mathcal{J} \to \mathcal{I}$ and therefore a map of $\mathcal{O}'$-modules
\[ (f')^*\mathcal{J} \longrightarrow \mathcal{I} \]
If $i$ and $t$ are first order thickenings, then $(f')^*\mathcal{J} = f^*\mathcal{J}$ and the map above becomes a map $f^*\mathcal{J} \to \mathcal{I}$.
Definition 91.9.2. In Situation 91.9.1 we say that $(f, f')$ is a strict morphism of thickenings if the map $(f')^*\mathcal{J} \longrightarrow \mathcal{I}$ is surjective.
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