Lemma 76.52.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$. The following are equivalent:

$E$ is $Y$-perfect,

for every commutative diagram

\[ \xymatrix{ U \ar[d] \ar[r]_ g & V \ar[d] \\ X \ar[r]^ f & Y } \]where $U$, $V$ are schemes and the vertical arrows are étale, the complex $E|_ U$ is $V$-perfect in the sense of Derived Categories of Schemes, Definition 36.35.1,

for some commutative diagram as in (2) with $U \to X$ surjective, the complex $E|_ U$ is $V$-perfect in the sense of Derived Categories of Schemes, Definition 36.35.1,

for every commutative diagram as in (2) with $U$ and $V$ affine the complex $R\Gamma (U, E)$ is $\mathcal{O}_ Y(V)$-perfect.

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