The Stacks project

Remark 98.20.2 (Strong effectiveness). Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume we have

1. an affine open $\mathop{\mathrm{Spec}}(\Lambda ) \subset S$,

2. an inverse system $(R_ n)$ of $\Lambda$-algebras with surjective transition maps whose kernels are locally nilpotent,

3. a system $(\xi _ n)$ of objects of $\mathcal{X}$ lying over the system $(\mathop{\mathrm{Spec}}(R_ n))$.

In this situation, set $R = \mathop{\mathrm{lim}}\nolimits R_ n$. We say that $(\xi _ n)$ is effective if there exists an object $\xi$ of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(R)$ whose restriction to $\mathop{\mathrm{Spec}}(R_ n)$ gives the system $(\xi _ n)$.