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.

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