The Stacks project

Lemma 57.15.2. With notation as in Definition 57.15.1 let $K$ be the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$. Then the corresponding Fourier-Mukai functors $\Phi _ K : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ (Lemma 57.8.2) and $\Phi _ K : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ (Lemma 57.8.4) are equivalences.

Proof. Immediate from Lemma 57.8.3 and Example 57.8.6. $\square$

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