The Stacks project

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 ├ętale 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 ( and (,

  6. the functor ( is an equivalence,

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

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