Section 98.14 (07XJ): Axioms—The Stacks project

98.14 Axioms

Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Here are the axioms we will consider on $\mathcal{X}$ .

a set theoretic condition ¹ to be ignored by readers who are not interested in set theoretical issues,
$\mathcal{X}$ is a stack in groupoids for the étale topology,
$\mathcal{X}$ is limit preserving,
$\mathcal{X}$ satisfies the Rim-Schlessinger condition (RS),
the spaces $T\mathcal{F}_{\mathcal{X}, k, x_0}$ and $\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0})$ are finite dimensional for every $k$ and $x_0$ , see (98.8.0.1) and (98.8.0.2),
the functor (98.9.3.1) is an equivalence,
$\mathcal{X}$ and $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ satisfy openness of versality.

[1] The condition is the following: the supremum of all the cardinalities

$|\mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_{\mathop{\mathrm{Spec}}(k)})/\cong |$ and

$|\text{Arrows}(\mathcal{X}_{\mathop{\mathrm{Spec}}(k)})|$ where

$k$ runs over the finite type fields over

$S$ is

$\leq$ than the size of some object of

$(\mathit{Sch}/S)_{fppf}$ .

Comments (0)

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).

98.14 Axioms

Comments (0)

Post a comment