Remark 114.20.1 (Direct construction). Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $U$ be another algebraic space over $B$. Denote $q : X \times _ B U \to U$ the second projection. Consider the distinguished triangle

of Cotangent, Section 91.28. For any sheaf $\mathcal{F}$ of $\mathcal{O}_{X \times _ B U}$-modules we have the Atiyah class

see Cotangent, Section 91.19. We can compose this with the map to $E$ and choose a distinguished triangle

in $D(\mathcal{O}_{X \times _ B U})$. By construction the Atiyah class lifts to a map

fitting into a morphism of distinguished triangles

Given $S, B, X, f, U, \mathcal{F}$ we fix a choice of $E(\mathcal{F})$ and $e_\mathcal {F}$.

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