The Stacks project

Lemma 98.17.1. Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Assume that

  1. $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces,

  2. $\mathcal{X}$ satisfies axioms [-1], [0], [1], [2], [3] (see Section 98.14),

  3. every formal object of $\mathcal{X}$ is effective,

  4. $\mathcal{X}$ satisfies openness of versality, and

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

Then $\mathcal{X}$ is an algebraic stack.

Proof. Lemma 98.13.8 applies to $\mathcal{X}$. Using this we choose, for every finite type field $k$ over $S$ and every isomorphism class of object $x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_{\mathop{\mathrm{Spec}}(k)})$, an affine scheme $U_{k, x_0}$ of finite type over $S$ and a smooth morphism $(\mathit{Sch}/U_{k, x_0})_{fppf} \to \mathcal{X}$ 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

\[ (\mathit{Sch}/U)_{fppf} \to \mathcal{X}, \quad \text{with}\quad U = \coprod \nolimits _{k, x_0} U_{k, x_0} \]

is smooth1. To finish the proof it suffices to show this map is surjective, see Criteria for Representability, Lemma 97.19.1 (this is where we use axiom [0]). By Criteria for Representability, Lemma 97.5.6 it suffices to show that $(\mathit{Sch}/U)_{fppf} \times _\mathcal {X} (\mathit{Sch}/V)_{fppf} \to (\mathit{Sch}/V)_{fppf}$ is surjective for those $y : (\mathit{Sch}/V)_{fppf} \to \mathcal{X}$ where $V$ is an affine scheme locally of finite presentation over $S$. By assumption (1) the fibre product $(\mathit{Sch}/U)_{fppf} \times _\mathcal {X} (\mathit{Sch}/V)_{fppf}$ is representable by an algebraic space $W$. Then $W \to V$ is smooth, hence the image is open. Hence it suffices to show that the image of $W \to V$ 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 $x_0 = y|_{\mathop{\mathrm{Spec}}(k)}$ the pullback of $y$ by $v_0$. Then $(u_{k, x_0}, v_0)$ will give a morphism $\mathop{\mathrm{Spec}}(k) \to W$ whose composition with $W \to V$ is $v_0$ 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)

There are also:

  • 2 comment(s) on Section 98.17: Algebraic stacks

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 07Y4. Beware of the difference between the letter 'O' and the digit '0'.