The Stacks project

Lemma 61.12.10. Let $S$ be a scheme. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site containing $S$. Let $\mathit{Sch}$ be the category of all schemes.

  1. The categories $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$, $S_{pro\text{-}\acute{e}tale}$, and $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ have fibre products agreeing with fibre products in $\mathit{Sch}$.

  2. The categories $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$, $S_{pro\text{-}\acute{e}tale}$ have equalizers agreeing with equalizers in $\mathit{Sch}$.

  3. The categories $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$, and $S_{pro\text{-}\acute{e}tale}$ both have a final object, namely $S/S$.

  4. The category $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ has a final object agreeing with the final object of $\mathit{Sch}$, namely $\mathop{\mathrm{Spec}}(\mathbf{Z})$.

Proof. The category $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ contains $\mathop{\mathrm{Spec}}(\mathbf{Z})$ and is closed under products and fibre products by construction, see Sets, Lemma 3.9.9. Suppose we have $U \to S$, $V \to U$, $W \to U$ morphisms of schemes with $U, V, W \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{pro\text{-}\acute{e}tale})$. The fibre product $V \times _ U W$ in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ 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 $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$. This proves the result for $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$. If $U \to S$, $V \to U$ and $W \to U$ are weakly étale then so is $V \times _ U W \to S$ (see More on Morphisms, Section 37.64) and hence we get fibre products for $S_{pro\text{-}\acute{e}tale}$. If $U, V, W$ are affine, so is $V \times _ U W$ and hence we get fibre products for $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$.

Let $a, b : U \to V$ be two morphisms in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$. In this case the equalizer of $a$ and $b$ (in the category of schemes) is

\[ V \times _{\Delta _{V/\mathop{\mathrm{Spec}}(\mathbf{Z})}, V \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} V, (a, b)} (U \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} U) \]

which is an object of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ by what we saw above. Thus $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ has equalizers. If $a$ and $b$ are morphisms over $S$, then the equalizer (in the category of schemes) is also given by

\[ V \times _{\Delta _{V/S}, V \times _ S V, (a, b)} (U \times _ S U) \]

hence we see that $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ has equalizers. Moreover, if $U$ and $V$ are weakly-étale over $S$, then so is the equalizer above as a fibre product of schemes weakly étale over $S$. Thus $S_{pro\text{-}\acute{e}tale}$ has equalizers. The statements on final objects is clear. $\square$


Comments (0)

There are also:

  • 6 comment(s) on Section 61.12: The pro-étale site

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 098M. Beware of the difference between the letter 'O' and the digit '0'.