The Stacks project

Lemma 20.44.5. 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

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

  2. 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 20.44.4. For the converse, note that taking stalks commutes with tensor products (Modules, Lemma 17.15.1). 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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