Definition 38.5.2. 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 complete dévissage of \mathcal{F}/X/S at x is given by a system
(Z_ k, Y_ k, i_ k, \pi _ k, \mathcal{G}_ k, \alpha _ k, z_ k, y_ k)_{k = 1, \ldots , n}
such that (Z_ k, Y_ k, i_ k, \pi _ k, \mathcal{G}_ k, \alpha _ k) is a complete dévissage of \mathcal{F}/X/S over s, and such that
(Z_1, Y_1, i_1, \pi _1, \mathcal{G}_1, z_1, y_1) is a one step dévissage of \mathcal{F}/X/S at x,
for k = 2, \ldots , n the system (Z_ k, Y_ k, i_ k, \pi _ k, \mathcal{G}_ k, z_ k, y_ k) is a one step dévissage of \mathop{\mathrm{Coker}}(\alpha _{k - 1})/Y_{k - 1}/S at y_{k - 1}.
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