The Stacks project

Lemma 21.44.10. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. Let $a, b \in \mathbf{Z}$.

  1. If $E$ has tor amplitude in $[a, b]$, then for every point $p$ of the site $\mathcal{C}$ the object $E_ p$ of $D(\mathcal{O}_ p)$ has tor amplitude in $[a, b]$.

  2. If $\mathcal{C}$ has enough points, then the converse is true.

Proof. Proof of (1). This follows because taking stalks at $p$ is the same as pulling back by the morphism of ringed sites $(p, \mathcal{O}_ p) \to (\mathcal{C}, \mathcal{O})$ and hence we can apply Lemma 21.44.5.

Proof of (2). If $\mathcal{C}$ has enough points, then we can check vanishing of $H^ i(E \otimes _\mathcal {O}^\mathbf {L} \mathcal{F})$ at stalks, see Modules on Sites, Lemma 18.14.4. Since $H^ i(E \otimes _\mathcal {O}^\mathbf {L} \mathcal{F})_ p = H^ i(E_ p \otimes _{\mathcal{O}_ p}^\mathbf {L} \mathcal{F}_ p)$ we conclude. $\square$


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