## 97.14 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 (97.8.0.1) and (97.8.0.2),

6. the functor (97.9.3.1) is an equivalence,

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

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

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