Definition 38.4.1. 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 $s \in S$ be a point. A one step dévissage of $\mathcal{F}/X/S$ over $s$ is given by morphisms of schemes over $S$

$\xymatrix{ X & Z \ar[l]_ i \ar[r]^\pi & Y }$

and a quasi-coherent $\mathcal{O}_ Z$-module $\mathcal{G}$ of finite type such that

1. $X$, $S$, $Z$ and $Y$ are affine,

2. $i$ is a closed immersion of finite presentation,

3. $\mathcal{F} \cong i_*\mathcal{G}$,

4. $\pi$ is finite, and

5. the structure morphism $Y \to S$ is smooth with geometrically irreducible fibres of dimension $\dim (\text{Supp}(\mathcal{F}_ s))$.

In this case we say $(Z, Y, i, \pi , \mathcal{G})$ is a one step dévissage of $\mathcal{F}/X/S$ over $s$.

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