The Stacks project

Definition 38.4.2. Let $S$ be a scheme. Let $X$ be locally of finite type over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. Let $x \in X$ be a point with image $s$ in $S$. A one step dévissage of $\mathcal{F}/X/S$ at $x$ is a system $(Z, Y, i, \pi , \mathcal{G}, z, y)$, where $(Z, Y, i, \pi , \mathcal{G})$ is a one step dévissage of $\mathcal{F}/X/S$ over $s$ and

1. $\dim _ x(\text{Supp}(\mathcal{F}_ s)) = \dim (\text{Supp}(\mathcal{F}_ s))$,

2. $z \in Z$ is a point with $i(z) = x$ and $\pi (z) = y$,

3. we have $\pi ^{-1}(\{ y\} ) = \{ z\} $,

4. the extension $\kappa (y)/\kappa (s)$ is purely transcendental.