The Stacks project

Lemma 34.3.8. Let $S$ be a scheme. Let $\mathit{Sch}_{Zar}$ be a big Zariski site containing $S$. Both $S_{Zar}$ and $(\textit{Aff}/S)_{Zar}$ are sites.

Proof. Let us show that $S_{Zar}$ is a site. It is a category with a given set of families of morphisms with fixed target. Thus we have to show properties (1), (2) and (3) of Sites, Definition 7.6.2. Since $(\mathit{Sch}/S)_{Zar}$ is a site, it suffices to prove that given any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{Zar}$ with $U \in \mathop{\mathrm{Ob}}\nolimits (S_{Zar})$ we also have $U_ i \in \mathop{\mathrm{Ob}}\nolimits (S_{Zar})$. This follows from the definitions as the composition of open immersions is an open immersion.

Let us show that $(\textit{Aff}/S)_{Zar}$ is a site. Reasoning as above, it suffices to show that the collection of standard Zariski coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 7.6.2. Let $R$ be a ring. Let $f_1, \ldots , f_ n \in R$ generate the unit ideal. For each $i \in \{ 1, \ldots , n\} $ let $g_{i1}, \ldots , g_{in_ i} \in R_{f_ i}$ be elements generating the unit ideal of $R_{f_ i}$. Write $g_{ij} = f_{ij}/f_ i^{e_{ij}}$ which is possible. After replacing $f_{ij}$ by $f_ i f_{ij}$ if necessary, we have that $D(f_{ij}) \subset D(f_ i) \cong \mathop{\mathrm{Spec}}(R_{f_ i})$ is equal to $D(g_{ij}) \subset \mathop{\mathrm{Spec}}(R_{f_ i})$. Hence we see that the family of morphisms $\{ D(g_{ij}) \to \mathop{\mathrm{Spec}}(R)\} $ is a standard Zariski covering. From these considerations it follows that (2) holds for standard Zariski coverings. We omit the verification of (1) and (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Your email address will not be published.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 020U. Beware of the difference between the letter 'O' and the digit '0'.