The Stacks project

Definition 21.44.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. Let $a, b \in \mathbf{Z}$ with $a \leq b$.

  1. We say $E$ has tor-amplitude in $[a, b]$ if $H^ i(E \otimes _\mathcal {O}^\mathbf {L} \mathcal{F}) = 0$ for all $\mathcal{O}$-modules $\mathcal{F}$ and all $i \not\in [a, b]$.

  2. We say $E$ has finite tor dimension if it has tor-amplitude in $[a, b]$ for some $a, b$.

  3. We say $E$ locally has finite tor dimension if for any object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} $ such that $E|_{U_ i}$ has finite tor dimension for all $i$.

An $\mathcal{O}$-module $\mathcal{F}$ has tor dimension $\leq d$ if $\mathcal{F}[0]$ viewed as an object of $D(\mathcal{O})$ has tor-amplitude in $[-d, 0]$.


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Your email address will not be published.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 08FZ. Beware of the difference between the letter 'O' and the digit '0'.