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
(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{F})_ x = E_ x \otimes _{\mathcal{O}_{X, x}}^\mathbf {L} \mathcal{F}_ x
On the other hand, taking stalks is exact, so
H^ i(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{F})_ x = H^ i((E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{F})_ x) = H^ i(E_ x \otimes _{\mathcal{O}_{X, x}}^\mathbf {L} \mathcal{F}_ x)
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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