The Stacks project

Lemma 75.13.7. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. For $m \in \mathbf{Z}$ the following are equivalent

  1. $H^ i(E)$ is coherent for $i \geq m$ and zero for $i \gg 0$, and

  2. $E$ is $m$-pseudo-coherent.

In particular, $E$ is pseudo-coherent if and only if $E$ is an object of $D^-_{\textit{Coh}}(\mathcal{O}_ X)$.

Proof. As $X$ is quasi-compact we can find an affine scheme $U$ and a surjective ├ętale morphism $U \to X$ (Properties of Spaces, Lemma 66.6.3). Observe that $U$ is Noetherian. Note that $E$ is $m$-pseudo-coherent if and only if $E|_ U$ is $m$-pseudo-coherent (follows from the definition or from Cohomology on Sites, Lemma 21.45.2). Similarly, $H^ i(E)$ is coherent if and only if $H^ i(E)|_ U = H^ i(E|_ U)$ is coherent (see Cohomology of Spaces, Lemma 69.12.2). Thus we may assume that $X$ is representable.

If $X$ is representable by a scheme $X_0$ then (Lemma 75.4.2) we can write $E = \epsilon ^*E_0$ where $E_0$ is an object of $D_\mathit{QCoh}(\mathcal{O}_{X_0})$ and $\epsilon : X_{\acute{e}tale}\to (X_0)_{Zar}$ is as in (75.4.0.1). In this case $E$ is $m$-pseudo-coherent if and only if $E_0$ is by Lemma 75.13.2. Similarly, $H^ i(E_0)$ is of finite type (i.e., coherent) if and only if $H^ i(E)$ is by Lemma 75.13.1. Finally, $H^ i(E_0) = 0$ if and only if $H^ i(E) = 0$ by Lemma 75.4.1. Thus we reduce to the case of schemes which is Derived Categories of Schemes, Lemma 36.10.3. $\square$


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