The Stacks project

Lemma 75.13.8. Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $a \leq b$. The following are equivalent

  1. $E$ has tor amplitude in $[a, b]$, and

  2. for all $\mathcal{F}$ in $\mathit{QCoh}(\mathcal{O}_ X)$ we have $H^ i(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{F}) = 0$ for $i \not\in [a, b]$.

Proof. It is clear that (1) implies (2). Assume (2). Let $j : U \to X$ be an ├ętale morphism with $U$ affine. As $X$ is quasi-separated $j : U \to X$ is quasi-compact and separated, hence $j_*$ transforms quasi-coherent modules into quasi-coherent modules (Morphisms of Spaces, Lemma 67.11.2). Thus the functor $\mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ U)$ is essentially surjective. It follows that condition (2) implies the vanishing of $H^ i(E|_ U \otimes _{\mathcal{O}_ U}^\mathbf {L} \mathcal{G})$ for $i \not\in [a, b]$ for all quasi-coherent $\mathcal{O}_ U$-modules $\mathcal{G}$. Since it suffices to prove that $E|_ U$ has tor amplitude in $[a, b]$ we reduce to the case where $X$ 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). For every quasi-coherent module $\mathcal{F}_0$ on $X_0$ the module $\epsilon ^*\mathcal{F}_0$ is quasi-coherent on $X$ and

\[ H^ i(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \epsilon ^*\mathcal{F}_0) = \epsilon ^*H^ i(E_0 \otimes _{\mathcal{O}_{X_0}}^\mathbf {L} \mathcal{F}_0) \]

as $\epsilon $ is flat (Lemma 75.4.1). Moreover, the vanishing of these sheaves for $i \not\in [a, b]$ implies the same thing for $H^ i(E_0 \otimes _{\mathcal{O}_{X_0}}^\mathbf {L} \mathcal{F}_0)$ by the same lemma. Thus we've reduced the problem to the case of schemes which is treated in Derived Categories of Schemes, Lemma 36.10.6. $\square$


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