The Stacks project

Proposition 97.17.2. 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. $\Delta _\Delta : \mathcal{X} \to \mathcal{X} \times _{\mathcal{X} \times \mathcal{X}} \mathcal{X}$ is representable by algebraic spaces,

  2. $\mathcal{X}$ satisfies axioms [-1], [0], [1], [2], [3], [4], and [5] (see Section 97.14),

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

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

Proof. We first prove that $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces. To do this it suffices to show that

\[ \mathcal{Y} = \mathcal{X} \times _{\Delta , \mathcal{X} \times \mathcal{X}, y} (\mathit{Sch}/V)_{fppf} \]

is representable by an algebraic space for any affine scheme $V$ locally of finite presentation over $S$ and object $y$ of $\mathcal{X} \times \mathcal{X}$ over $V$, see Criteria for Representability, Lemma 96.5.51. Observe that $\mathcal{Y}$ is fibred in setoids (Stacks, Lemma 8.2.5) and let $Y : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$, $T \mapsto \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_ T)/\cong $ be the functor of isomorphism classes. We will apply Proposition 97.16.1 to see that $Y$ is an algebraic space.

Note that $\Delta _\mathcal {Y} : \mathcal{Y} \to \mathcal{Y} \times \mathcal{Y}$ (and hence also $Y \to Y \times Y$) is representable by algebraic spaces by condition (1) and Criteria for Representability, Lemma 96.4.4. Observe that $Y$ is a sheaf for the ├ętale topology by Stacks, Lemmas 8.6.3 and 8.6.7, i.e., axiom [0] holds. Also $Y$ is limit preserving by Lemma 97.11.2, i.e., we have [1]. Note that $Y$ has (RS), i.e., axiom [2] holds, by Lemmas 97.5.2 and 97.5.3. Axiom [3] for $Y$ follows from Lemmas 97.8.1 and 97.8.2. Axiom [4] follows from Lemmas 97.9.5 and 97.9.6. Axiom [5] for $Y$ follows directly from openness of versality for $\Delta _\mathcal {X}$ which is part of axiom [5] for $\mathcal{X}$. Thus all the assumptions of Proposition 97.16.1 are satisfied and $Y$ is an algebraic space.

At this point it follows from Lemma 97.17.1 that $\mathcal{X}$ is an algebraic stack. $\square$

[1] The set theoretic condition in Criteria for Representability, Lemma 96.5.5 will hold: the size of the algebraic space $Y$ representing $\mathcal{Y}$ is suitably bounded. Namely, $Y \to S$ will be locally of finite type and $Y$ will satisfy axiom [-1]. Details omitted.

Comments (0)

There are also:

  • 2 comment(s) on Section 97.17: 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 07Y5. Beware of the difference between the letter 'O' and the digit '0'.