Lemma 36.35.3. Let f : X \to S be a morphism of schemes which is flat and locally of finite presentation. Let E be an object of D_\mathit{QCoh}(\mathcal{O}_ X). The following are equivalent
E is S-perfect,
for any affine open U \subset X mapping into an affine open V \subset S the complex R\Gamma (U, E) is \mathcal{O}_ S(V)-perfect.
there exists an affine open covering S = \bigcup V_ i and for each i an affine open covering f^{-1}(V_ i) = \bigcup U_{ij} such that the complex R\Gamma (U_{ij}, E) is \mathcal{O}_ S(V_ i)-perfect.
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