Lemma 20.48.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $E$ be an object of $D(\mathcal{O}_ X)$. Let $a, b \in \mathbf{Z}$ with $a \leq b$. The following are equivalent
$E$ has tor-amplitude in $[a, b]$.
for every $x \in X$ the object $E_ x$ of $D(\mathcal{O}_{X, x})$ has tor-amplitude in $[a, b]$.
Proof. Taking stalks at $x$ is the same thing as pulling back by the morphism of ringed spaces $(x, \mathcal{O}_{X, x}) \to (X, \mathcal{O}_ X)$. Hence the implication (1) $\Rightarrow $ (2) follows from Lemma 20.48.4. For the converse, note that taking stalks commutes with tensor products (Modules, Lemma 17.16.1). Hence
On the other hand, taking stalks is exact, so
and we can check whether $H^ i(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{F})$ is zero by checking whether all of its stalks are zero (Modules, Lemma 17.3.1). Thus (2) implies (1). $\square$
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