Lemma 37.59.7. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Fix $m \in \mathbf{Z}$. The following are equivalent

$E$ is $m$-pseudo-coherent relative to $S$,

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 of $\mathcal{O}_ X(U_{ij})$-modules $R\Gamma (U_{ij}, E)$ is $m$-pseudo-coherent relative to $\mathcal{O}_ S(V_ i)$, and

for every affine opens $U \subset X$ and $V \subset S$ with $f(U) \subset V$ the complex of $\mathcal{O}_ X(U)$-modules $R\Gamma (U, E)$ is $m$-pseudo-coherent relative to $\mathcal{O}_ S(V)$.

## Comments (0)