The Stacks project

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

  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 20.48.4. 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). $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09U9. Beware of the difference between the letter 'O' and the digit '0'.