The Stacks project

Lemma 91.16.6. Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}_0$ be a surjection of sheaves of rings whose kernel is an ideal sheaf $\mathcal{I}$ of square zero. Let $K, L \in D^-(\mathcal{O})$. Set $K_0 = K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}_0$ and $L_0 = L \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}_0$ in $D^-(\mathcal{O}_0)$. Given $\alpha _0 : K_0 \to L_0$ in $D(\mathcal{O}_0)$ there is a canonical element

\[ o(\alpha _0) \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_0}(K_0, L_0 \otimes _{\mathcal{O}_0}^\mathbf {L} \mathcal{I}) \]

whose vanishing is necessary and sufficient for the existence of a map $\alpha : K \to L$ in $D(\mathcal{O})$ with $\alpha _0 = \alpha \otimes _\mathcal {O}^\mathbf {L} \text{id}$.

Proof. Finding $\alpha : K \to L$ lifing $\alpha _0$ is the same as finding $\alpha : K \to L$ such that the composition $K \xrightarrow {\alpha } L \to L_0$ is equal to the composition $K \to K_0 \xrightarrow {\alpha _0} L_0$. The short exact sequence $0 \to \mathcal{I} \to \mathcal{O} \to \mathcal{O}_0 \to 0$ gives rise to a canonical distinguished triangle

\[ L \otimes _\mathcal {O}^\mathbf {L} \mathcal{I} \to L \to L_0 \to (L \otimes _\mathcal {O}^\mathbf {L} \mathcal{I})[1] \]

in $D(\mathcal{O})$. By Derived Categories, Lemma 13.4.2 the composition

\[ K \to K_0 \xrightarrow {\alpha _0} L_0 \to (L \otimes _\mathcal {O}^\mathbf {L} \mathcal{I})[1] \]

is zero if and only if we can find $\alpha : K \to L$ lifting $\alpha _0$. The composition is an element in

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(K, (L \otimes _\mathcal {O}^\mathbf {L} \mathcal{I})[1]) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_0)}(K_0, (L \otimes _\mathcal {O}^\mathbf {L} \mathcal{I})[1]) = \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_0}(K_0, L_0 \otimes _{\mathcal{O}_0}^\mathbf {L} \mathcal{I}) \]

by adjunction. $\square$


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