The Stacks project

Lemma 36.10.6. Let $X$ be a quasi-separated scheme. 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 $U \subset X$ be an affine open. As $X$ is quasi-separated the morphism $j : U \to X$ is quasi-compact and separated, hence $j_*$ transforms quasi-coherent modules into quasi-coherent modules (Schemes, Lemma 26.24.1). 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 $\mathcal{G}$. Write $U = \mathop{\mathrm{Spec}}(A)$ and let $M^\bullet $ be the complex of $A$-modules corresponding to $E|_ U$ by Lemma 36.3.5. We have just shown that $M^\bullet \otimes _ A^\mathbf {L} N$ has vanishing cohomology groups outside the range $[a, b]$, in other words $M^\bullet $ has tor amplitude in $[a, b]$. By Lemma 36.10.4 we conclude that $E|_ U$ has tor amplitude in $[a, b]$. This proves the lemma. $\square$

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