Lemma 91.14.4. In the situation above assume that X is quasi-compact and quasi-separated and that DQ_ X(\mathcal{F}) \to DQ_ X(\mathcal{G}) (Derived Categories of Spaces, Section 75.19) is an isomorphism. Then the functor FT is an equivalence of categories.
Proof. A solution of (91.13.0.1) for \mathcal{F} in particular gives an extension of f^{-1}\mathcal{O}_{B'}-algebras
0 \to \mathcal{F} \to \mathcal{O}' \to \mathcal{O}_ X \to 0
where \mathcal{F} is an ideal of square zero. Similarly for \mathcal{G}. Moreover, given such an extension, we obtain a map c_{\mathcal{O}'} : f^{-1}\mathcal{J} \to \mathcal{F}. Thus we are looking at the full subcategory of such extensions of f^{-1}\mathcal{O}_{B'}-algebras with c = c_{\mathcal{O}'}. Clearly, if \mathcal{O}'' = F(\mathcal{O}') where F is the equivalence of Lemma 91.14.3 (applied to X \to B' this time), then c_{\mathcal{O}''} is the composition of c_{\mathcal{O}'} and the map \mathcal{F} \to \mathcal{G}. This proves the lemma. \square
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