The Stacks project

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

\[ Lq^*L_{U/B} \to L_{X \times _ B U/B} \to E \to Lq^*L_{U/B}[1] \]

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

\[ \mathcal{F} \to L_{X \times _ B U/B} \otimes _{\mathcal{O}_{X \times _ B U}}^\mathbf {L} \mathcal{F}[1] \]

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

\[ E(\mathcal{F}) \to \mathcal{F} \to \mathcal{F} \otimes _{\mathcal{O}_{X \times _ B U}}^\mathbf {L} E[1] \to E(\mathcal{F})[1] \]

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

\[ e_\mathcal {F} : E(\mathcal{F}) \longrightarrow Lq^*L_{U/B} \otimes _{\mathcal{O}_{X \times _ B U}}^\mathbf {L} \mathcal{F}[1] \]

fitting into a morphism of distinguished triangles

\[ \xymatrix{ \mathcal{F} \otimes ^\mathbf {L} Lq^*L_{U/B}[1] \ar[r] & \mathcal{F} \otimes ^\mathbf {L} L_{X \times _ B U/B}[1] \ar[r] & \mathcal{F} \otimes ^\mathbf {L} E[1] \\ E(\mathcal{F}) \ar[r] \ar[u]^{e_\mathcal {F}} & \mathcal{F} \ar[r] \ar[u]^{Atiyah} & \mathcal{F} \otimes ^\mathbf {L} E[1] \ar[u]^{=} } \]

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

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