Lemma 75.23.2 (08JN)—The Stacks project

Lemma 75.23.2. Assumption and notation as in Lemma 75.22.3. Then there are functorial isomorphisms

\[ H^ i(B, K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) \]

for $\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps (see proof).

Proof. As in the proof of Lemma 75.22.3 let $E^\vee $ be the dual perfect complex and recall that $K = Rf_*(E^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{G}^\bullet )$. Since we also have

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) = H^ i(X, E^\vee \otimes ^\mathbf {L}_{\mathcal{O}_ X} (\mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F})) \]

by construction of $E^\vee $, the existence of the isomorphisms follows from Lemma 75.23.1 applied to $E^\vee $ and $\mathcal{G}^\bullet $. The statement on boundary maps means the following: Given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ then the isomorphisms fit into commutative diagrams

\[ \xymatrix{ H^ i(B, K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_3) \ar[r] \ar[d]_\delta & \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_3) \ar[d]^\delta \\ H^{i + 1}(B, K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_1) \ar[r] & \mathop{\mathrm{Ext}}\nolimits ^{i + 1}_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_1) } \]

where the boundary maps come from the distinguished triangle

\[ K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_1 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_2 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_3 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_1[1] \]

and the distinguished triangle in $D(\mathcal{O}_ X)$ associated to the short exact sequence

\[ 0 \to \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_1 \to \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_2 \to \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_3 \to 0 \]

of complexes. This sequence is exact because $\mathcal{G}^ n$ is flat over $B$. We omit the verification of the commutativity of the displayed diagram. $\square$

The Stacks project

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