Lemma 91.6.3. In Situation 91.6.2 the modules $\pi ^*\mathcal{F}$ and $h^*\mathcal{F}'_2$ are $\mathcal{O}'_1$-modules flat over $S'_1$ restricting to $\mathcal{F}$ on $X$. Their difference (Lemma 91.6.1) is an element $\theta $ of $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}( \mathcal{F}, f^*\mathcal{J}_1 \otimes _{\mathcal{O}_ X} \mathcal{F})$ whose boundary in $\mathop{\mathrm{Ext}}\nolimits ^2_{\mathcal{O}_ X}( \mathcal{F}, f^*\mathcal{J}_3 \otimes _{\mathcal{O}_ X} \mathcal{F})$ equals the obstruction (Lemma 91.6.1) to lifting $\mathcal{F}$ to an $\mathcal{O}'_3$-module flat over $S'_3$.
Proof. Note that both $\pi ^*\mathcal{F}$ and $h^*\mathcal{F}'_2$ restrict to $\mathcal{F}$ on $X$ and that the kernels of $\pi ^*\mathcal{F} \to \mathcal{F}$ and $h^*\mathcal{F}'_2 \to \mathcal{F}$ are given by $f^*\mathcal{J}_1 \otimes _{\mathcal{O}_ X} \mathcal{F}$. Hence flatness by Lemma 91.5.2. Taking the boundary makes sense as the sequence of modules
\[ 0 \to f^*\mathcal{J}_3 \otimes _{\mathcal{O}_ X} \mathcal{F} \to f^*\mathcal{J}_2 \otimes _{\mathcal{O}_ X} \mathcal{F} \to f^*\mathcal{J}_1 \otimes _{\mathcal{O}_ X} \mathcal{F} \to 0 \]
is short exact due to the assumptions in Situation 91.6.2 and the fact that $\mathcal{F}$ is flat over $S$. The statement on the obstruction class is a direct translation of the result of Remark 91.4.10 to this particular situation. $\square$
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