The Stacks project

Remark 98.20.4. There is a way to deduce openness of versality of the diagonal of an category fibred in groupoids from a strong formal effectiveness axiom. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume

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

2. $\mathcal{X}$ has (RS*),

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

4. given an inverse system $(R_ n)$ of $S$-algebras as in Remark 98.20.2 where $\mathop{\mathrm{Ker}}(R_ m \to R_ n)$ is an ideal of square zero for all $m \geq n$ the functor

   \[ \mathcal{X}_{\mathop{\mathrm{Spec}}(\mathop{\mathrm{lim}}\nolimits R_ n)} \longrightarrow \mathop{\mathrm{lim}}\nolimits _ n \mathcal{X}_{\mathop{\mathrm{Spec}}(R_ n)} \]

   is fully faithful.

Then $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ satisfies openness of versality. This follows by applying Lemma 98.20.3 to fibre products of the form $\mathcal{X} \times _{\Delta , \mathcal{X} \times \mathcal{X}, y} (\mathit{Sch}/V)_{fppf}$ for any affine scheme $V$ locally of finite presentation over $S$ and object $y$ of $\mathcal{X} \times \mathcal{X}$ over $V$. If we ever need this, we will change this remark into a lemma and provide a detailed proof.