Loading web-font TeX/Math/Italic

The Stacks project

Definition 38.16.1. Let f : X \to S be a morphism of schemes which is of finite type. Let \mathcal{F} be a finite type quasi-coherent \mathcal{O}_ X-module.

  1. Let s \in S. We say \mathcal{F} is pure along X_ s if there is no impurity (g : T \to S, t' \leadsto t, \xi ) of \mathcal{F} above s with (T, t) \to (S, s) an elementary étale neighbourhood.

  2. We say \mathcal{F} is universally pure along X_ s if there does not exist any impurity of \mathcal{F} above s.

  3. We say that X is pure along X_ s if \mathcal{O}_ X is pure along X_ s.

  4. We say \mathcal{F} is universally S-pure, or universally pure relative to S if \mathcal{F} is universally pure along X_ s for every s \in S.

  5. We say \mathcal{F} is S-pure, or pure relative to S if \mathcal{F} is pure along X_ s for every s \in S.

  6. We say that X is S-pure or pure relative to S if \mathcal{O}_ X is pure relative to S.


Comments (0)


Your email address will not be published.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.