The Stacks project

Lemma 91.12.3. In Situation 91.12.2 the obstruction class $o(\mathcal{F}, f^*\mathcal{J}_2 \otimes _\mathcal {O} \mathcal{F}, 1)$ maps to the obstruction class $o(\mathcal{F}, f^*\mathcal{J}_1 \otimes _\mathcal {O} \mathcal{F}, 1)$ under the canonical map

\[ \mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {O}( \mathcal{F}, f^*\mathcal{J}_2 \otimes _\mathcal {O} \mathcal{F}) \to \mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {O}( \mathcal{F}, f^*\mathcal{J}_1 \otimes _\mathcal {O} \mathcal{F}) \]

Proof. Follows from Remark 91.10.9. $\square$

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