The Stacks project

Situation 91.12.4. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}), \mathcal{O}_\mathcal {B})$ be a morphism of ringed topoi. Consider a commutative diagram

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_1), \mathcal{O}'_1) \ar[r]_ h \ar[d]_{f'_1} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_2), \mathcal{O}'_2) \ar[r] \ar[d]_{f'_2} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_3), \mathcal{O}'_3) \ar[d]_{f'_3} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}'_1), \mathcal{O}_{\mathcal{B}'_1}) \ar[r] & (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}'_2), \mathcal{O}_{\mathcal{B}'_2}) \ar[r] & (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}'_3), \mathcal{O}_{\mathcal{B}'_3}) } \]

where (a) the top row is a short exact sequence of first order thickenings of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$, (b) the lower row is a short exact sequence of first order thickenings of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{B}), \mathcal{O}_\mathcal {B})$, (c) each $f'_ i$ restricts to $f$, (d) each pair $(f, f_ i')$ is a strict morphism of thickenings, and (e) each $f'_ i$ is flat. Finally, let $\mathcal{F}'_2$ be an $\mathcal{O}'_2$-module flat over $\mathcal{O}_{\mathcal{B}'_2}$ and set $\mathcal{F} = \mathcal{F}'_2 \otimes \mathcal{O}$. Let $\pi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be the canonical splitting (Remark 91.10.10).

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