Remark 36.35.12. The reader may have noticed the similarity between Lemma 36.35.11 and Lemma 36.28.3. Indeed, the pseudo-coherent complex $L$ of Lemma 36.35.11 may be characterized as the unique pseudo-coherent complex on $S$ such that there are functorial isomorphisms

$\mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ S}(L, \mathcal{F}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(K, E \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*\mathcal{F})$

compatible with boundary maps for $\mathcal{F}$ ranging over $\mathit{QCoh}(\mathcal{O}_ S)$. If we ever need this we will formulate a precise result here and give a detailed proof.

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