Lemma 34.3.9. Let $S$ be a scheme. Let $\mathit{Sch}_{Zar}$ be a big Zariski site containing $S$. The underlying categories of the sites $\mathit{Sch}_{Zar}$, $(\mathit{Sch}/S)_{Zar}$, $S_{Zar}$, $(\textit{Aff}/S)_{Zar}$, and $S_{affine, Zar}$ have fibre products. In each case the obvious functor into the category $\mathit{Sch}$ of all schemes commutes with taking fibre products. The categories $(\mathit{Sch}/S)_{Zar}$, and $S_{Zar}$ both have a final object, namely $S/S$.
Proof. For $\mathit{Sch}_{Zar}$ it is true by construction, see Sets, Lemma 3.9.9. Suppose we have $U \to S$, $V \to U$, $W \to U$ morphisms of schemes with $U, V, W \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{Zar})$. The fibre product $V \times _ U W$ in $\mathit{Sch}_{Zar}$ is a fibre product in $\mathit{Sch}$ and is the fibre product of $V/S$ with $W/S$ over $U/S$ in the category of all schemes over $S$, and hence also a fibre product in $(\mathit{Sch}/S)_{Zar}$. This proves the result for $(\mathit{Sch}/S)_{Zar}$. If $U \to S$, $V \to U$ and $W \to U$ are open immersions then so is $V \times _ U W \to S$ and hence we get the result for $S_{Zar}$. If $U, V, W$ are affine, so is $V \times _ U W$ and hence the result for $(\textit{Aff}/S)_{Zar}$ and $S_{affine, Zar}$. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: