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 $\mathcal{X}$.

1. a set theoretic condition¹ to be ignored by readers who are not interested in set theoretical issues,

2. $\mathcal{X}$ is a stack in groupoids for the étale topology,

3. $\mathcal{X}$ is limit preserving,

4. $\mathcal{X}$ satisfies the Rim-Schlessinger condition (RS),

5. 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),

6. the functor (98.9.3.1) is an equivalence,

7. $\mathcal{X}$ and $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ satisfy openness of versality.