The Stacks project

Remark 114.20.2 (Construction of obstruction class). With notation as in Remark 114.20.1 let $i : U \to U'$ be a first order thickening of $U$ over $B$. Let $\mathcal{I} \subset \mathcal{O}_{U'}$ be the quasi-coherent sheaf of ideals cutting out $B$ in $B'$. The fundamental triangle

\[ Li^*L_{U'/B} \to L_{U/B} \to L_{U/U'} \to Li^*L_{U'/B}[1] \]

together with the map $L_{U/U'} \to \mathcal{I}[1]$ determine a map $e_{U'} : L_{U/B} \to \mathcal{I}[1]$. Combined with the map $e_\mathcal {F}$ of the previous remark we obtain

\[ (\text{id}_\mathcal {F} \otimes Lq^*e_{U'}) \cup e_\mathcal {F} : E(\mathcal{F}) \longrightarrow \mathcal{F} \otimes _{\mathcal{O}_{X \times _ B U}} q^*\mathcal{I}[2] \]

(we have also composed with the map from the derived tensor product to the usual tensor product). In other words, we obtain an element

\[ \xi _{U'} \in \mathop{\mathrm{Ext}}\nolimits ^2_{\mathcal{O}_{X \times _ B U}}( E(\mathcal{F}), \mathcal{F} \otimes _{\mathcal{O}_{X \times _ B U}} q^*\mathcal{I}) \]

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