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
and a quasi-coherent \mathcal{O}_ Z-module \mathcal{G} of finite type such that
X, S, Z and Y are affine,
i is a closed immersion of finite presentation,
\mathcal{F} \cong i_*\mathcal{G},
\pi is finite, and
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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