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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