The Stacks project

Situation 91.6.2. Let $f : (X, \mathcal{O}_ X) \to (S, \mathcal{O}_ S)$ be a morphism of ringed spaces. Consider a commutative diagram

\[ \xymatrix{ (X'_1, \mathcal{O}'_1) \ar[r]_ h \ar[d]_{f'_1} & (X'_2, \mathcal{O}'_2) \ar[r] \ar[d]_{f'_2} & (X'_3, \mathcal{O}'_3) \ar[d]_{f'_3} \\ (S'_1, \mathcal{O}_{S'_1}) \ar[r] & (S'_2, \mathcal{O}_{S'_2}) \ar[r] & (S'_3, \mathcal{O}_{S'_3}) } \]

where (a) the top row is a short exact sequence of first order thickenings of $X$, (b) the lower row is a short exact sequence of first order thickenings of $S$, (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 $S'_2$ and set $\mathcal{F} = \mathcal{F}'_2|_ X$. Let $\pi : X'_1 \to X$ be the canonical splitting (Remark 91.4.9).

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View Situation 91.6.2 as pdf