The Stacks project

Lemma 28.20.3. Let $X$ be a reduced scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then the equivalent conditions of Lemma 28.20.2 are also equivalent to

  1. $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite type and the function

    \[ \rho _\mathcal {F} : X \to \mathbf{Z}, \quad x \longmapsto \dim _{\kappa (x)} \mathcal{F}_ x \otimes _{\mathcal{O}_{X, x}} \kappa (x) \]

    is locally constant in the Zariski topology on $X$.

Proof. This lemma immediately reduces to the affine case. In this case the lemma is a reformulation of Algebra, Lemma 10.77.3. $\square$


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Your email address will not be published.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