Remark 91.10.10. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. We define a sequence of morphisms of first order thickenings
of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ to be a complex if the corresponding maps between the ideal sheaves $\mathcal{I}_ i$ give a complex of $\mathcal{O}$-modules $\mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1$ (i.e., the composition is zero). In this case the composition $(\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_3), \mathcal{O}'_3)$ factors through $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_3), \mathcal{O}'_3)$, i.e., the first order thickening $(\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1)$ of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ is trivial and comes with a canonical trivialization $\pi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$.
We say a sequence of morphisms of first order thickenings
of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ is a short exact sequence if the corresponding maps between ideal sheaves is a short exact sequence
of $\mathcal{O}$-modules.
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