Lemma 36.28.2. Assumptions and notation as in Lemma 36.27.3. Then there are functorial isomorphisms
H^ i(S, K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \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 S compatible with boundary maps (see proof).
Proof.
As in the proof of Lemma 36.27.3 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 36.28.1 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(S, K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \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}(S, K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \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}_ S} \mathcal{F}_1 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_2 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_3 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \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} is flat over S. We omit the verification of the commutativity of the displayed diagram.
\square
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