The Stacks project

Proposition 98.16.1. Let $S$ be a locally Noetherian scheme. Let $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Assume that

  1. $\Delta : F \to F \times F$ is representable by algebraic spaces,

  2. $F$ satisfies axioms [-1], [0], [1], [2], [3], [4], [5] (see Section 98.15), and

  3. $\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$.

Then $F$ is an algebraic space.

Proof. Lemma 98.13.8 applies to $F$. Using this we choose, for every finite type field $k$ over $S$ and $x_0 \in F(\mathop{\mathrm{Spec}}(k))$, an affine scheme $U_{k, x_0}$ of finite type over $S$ and a smooth morphism $U_{k, x_0} \to F$ such that there exists a finite type point $u_{k, x_0} \in U_{k, x_0}$ with residue field $k$ such that $x_0$ is the image of $u_{k, x_0}$. Then

\[ U = \coprod \nolimits _{k, x_0} U_{k, x_0} \longrightarrow F \]

is smooth1. To finish the proof it suffices to show this map is surjective, see Bootstrap, Lemma 80.12.3 (this is where we use axiom [0]). By Criteria for Representability, Lemma 97.5.6 it suffices to show that $U \times _ F V \to V$ is surjective for those $V \to F$ where $V$ is an affine scheme locally of finite presentation over $S$. Since $U \times _ F V \to V$ is smooth the image is open. Hence it suffices to show that the image of $U \times _ F V \to V$ contains all finite type points of $V$, see Morphisms, Lemma 29.16.7. Let $v_0 \in V$ be a finite type point. Then $k = \kappa (v_0)$ is a finite type field over $S$. Denote $x_0$ the composition $\mathop{\mathrm{Spec}}(k) \xrightarrow {v_0} V \to F$. Then $(u_{k, x_0}, v_0) : \mathop{\mathrm{Spec}}(k) \to U \times _ F V$ is a point mapping to $v_0$ and we win. $\square$

[1] Set theoretical remark: This coproduct is (isomorphic) to an object of $(\mathit{Sch}/S)_{fppf}$ as we have a bound on the index set by axiom [-1], see Sets, Lemma 3.9.9.

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Your email address will not be published.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 07Y1. Beware of the difference between the letter 'O' and the digit '0'.