The Stacks project

This is discussed in [Schroeer].

Remark 59.30.2. Let $S = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. Let $(p, u)$ be a point of the site $(\textit{Aff}/S)_{fppf}$, see Sites, Sections 7.32 and 7.33. Let $B = \mathcal{O}_ p$ be the stalk of the structure sheaf at the point $p$. Recall that

\[ B = \mathop{\mathrm{colim}}\nolimits _{(U, x)} \mathcal{O}(U) = \mathop{\mathrm{colim}}\nolimits _{(\mathop{\mathrm{Spec}}(C), x_ C)} C \]

where $x_ C \in u(\mathop{\mathrm{Spec}}(C))$. It can happen that $\mathop{\mathrm{Spec}}(B)$ is an object of $(\textit{Aff}/S)_{fppf}$ and that there is an element $x_ B \in u(\mathop{\mathrm{Spec}}(B))$ mapping to the compatible system $x_ C$. In this case the system of neighbourhoods has an initial object and it follows that $\mathcal{F}_ p = \mathcal{F}(\mathop{\mathrm{Spec}}(B))$ for any sheaf $\mathcal{F}$ on $(\textit{Aff}/S)_{fppf}$. It is straightforward to see that if $\mathcal{F} \mapsto \mathcal{F}(\mathop{\mathrm{Spec}}(B))$ defines a point of $\mathop{\mathit{Sh}}\nolimits ((\textit{Aff}/S)_{fppf})$, then $B$ has to be a local $A$-algebra such that for every faithfully flat, finitely presented ring map $B \to B'$ there is a section $B' \to B$. Conversely, for any such $A$-algebra $B$ the functor $\mathcal{F} \mapsto \mathcal{F}(\mathop{\mathrm{Spec}}(B))$ is the stalk functor of a point. Details omitted. It is not clear what a general point of the site $(\textit{Aff}/S)_{fppf}$ looks like.

Comments (0)

There are also:

  • 2 comment(s) on Section 59.30: Points in other topologies

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Your email address will not be published.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 06VY. Beware of the difference between the letter 'O' and the digit '0'.