The Stacks project

Lemma 21.47.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. The following are equivalent

  1. $E$ is perfect, and

  2. $E$ is pseudo-coherent and locally has finite tor dimension.

Proof. Assume (1). Let $U$ be an object of $\mathcal{C}$. By definition there exists a covering $\{ U_ i \to U\} $ such that $E|_{U_ i}$ is represented by a strictly perfect complex. Thus $E$ is pseudo-coherent (i.e., $m$-pseudo-coherent for all $m$) by Lemma 21.45.2. Moreover, a direct summand of a finite free module is flat, hence $E|_{U_ i}$ has finite Tor dimension by Lemma 21.46.3. Thus (2) holds.

Assume (2). Let $U$ be an object of $\mathcal{C}$. After replacing $U$ by the members of a covering we may assume there exist integers $a \leq b$ such that $E|_ U$ has tor amplitude in $[a, b]$. Since $E|_ U$ is $m$-pseudo-coherent for all $m$ we conclude using Lemma 21.47.3. $\square$

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Your email address will not be published.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 08G8. Beware of the difference between the letter 'O' and the digit '0'.