The Stacks project

Definition 34.3.7. Let $S$ be a scheme. Let $\mathit{Sch}_{Zar}$ be a big Zariski site containing $S$.

  1. The big Zariski site of $S$, denoted $(\mathit{Sch}/S)_{Zar}$, is the site $\mathit{Sch}_{Zar}/S$ introduced in Sites, Section 7.25.

  2. The small Zariski site of $S$, which we denote $S_{Zar}$, is the full subcategory of $(\mathit{Sch}/S)_{Zar}$ whose objects are those $U/S$ such that $U \to S$ is an open immersion. A covering of $S_{Zar}$ is any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{Zar}$ with $U \in \mathop{\mathrm{Ob}}\nolimits (S_{Zar})$.

  3. The big affine Zariski site of $S$, denoted $(\textit{Aff}/S)_{Zar}$, is the full subcategory of $(\mathit{Sch}/S)_{Zar}$ consisting of objects $U/S$ such that $U$ is an affine scheme. A covering of $(\textit{Aff}/S)_{Zar}$ is any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{Zar}$ with $U \in \mathop{\mathrm{Ob}}\nolimits ((\textit{Aff}/S)_{Zar})$ which is a standard Zariski covering.

  4. The small affine Zariski site of $S$, denoted $S_{affine, Zar}$, is the full subcategory of $S_{Zar}$ whose objects are those $U/S$ such that $U$ is an affine scheme. A covering of $S_{affine, Zar}$ is any covering $\{ U_ i \to U\} $ of $S_{Zar}$ with $U \in \mathop{\mathrm{Ob}}\nolimits (S_{affine, Zar})$ which is a standard Zariski covering.

Comments (0)

There are also:

  • 2 comment(s) on Section 34.3: The Zariski topology

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Your email address will not be published.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 020T. Beware of the difference between the letter 'O' and the digit '0'.



View Definition 34.3.7 as pdf