The Stacks project

Lemma 37.57.7. 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

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

  2. 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

  3. 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 $\mathcal{O}_ S(V)$.

Proof. Let $U$ and $V$ be as in (2) and choose a closed immersion $i : U \to \mathbf{A}^ n_ V$. A formal argument, using Lemma 37.57.6, shows it suffices to prove that $Ri_*(E|_ U)$ is $m$-pseudo-coherent if and only if $R\Gamma (U, E)$ is $m$-pseudo-coherent relative to $\mathcal{O}_ S(V)$. Say $U = \mathop{\mathrm{Spec}}(A)$, $V = \mathop{\mathrm{Spec}}(R)$, and $\mathbf{A}^ n_ V = \mathop{\mathrm{Spec}}(R[x_1, \ldots , x_ n]$. By the remarks preceding the lemma, $E|_ U$ is quasi-isomorphic to the complex of quasi-coherent sheaves on $U$ associated to the object $R\Gamma (U, E)$ of $D(A)$. Note that $R\Gamma (U, E) = R\Gamma (\mathbf{A}^ n_ V, Ri_*(E|_ U))$ as $i$ is a closed immersion (and hence $i_*$ is exact). Thus $Ri_*E$ is associated to $R\Gamma (U, E)$ viewed as an object of $D(R[x_1, \ldots , x_ n])$. We conclude as $m$-pseudo-coherence of $Ri_*(E|_ U)$ is equivalent to $m$-pseudo-coherence of $R\Gamma (U, E)$ in $D(R[x_1, \ldots , x_ n])$ by Derived Categories of Schemes, Lemma 36.10.2 which is equivalent to $R\Gamma (U, E)$ is $m$-pseudo-coherent relative to $R = \mathcal{O}_ S(V)$ by definition. $\square$

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