Situation 91.9.1 (08M7)—The Stacks project

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 $f^*\mathcal{J} \to \mathcal{I}$.

The Stacks project

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