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

1. $(Z_1, Y_1, i_1, \pi _1, \mathcal{G}_1, z_1, y_1)$ is a one step dévissage of $\mathcal{F}/X/S$ at $x$,

2. 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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