Definition 38.5.1. 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 complete dévissage of $\mathcal{F}/X/S$ over $s$ is given by a diagram

$\xymatrix{ X & Z_1 \ar[l]^{i_1} \ar[d]^{\pi _1} \\ & Y_1 & Z_2 \ar[l]^{i_2} \ar[d]^{\pi _2} \\ & & Y_2 & Z_3 \ar[l] \ar[d] \\ & & & ... & ... \ar[l] \ar[d] \\ & & & & Y_ n }$

of schemes over $S$, finite type quasi-coherent $\mathcal{O}_{Z_ k}$-modules $\mathcal{G}_ k$, and $\mathcal{O}_{Y_ k}$-module maps

$\alpha _ k : \mathcal{O}_{Y_ k}^{\oplus r_ k} \longrightarrow \pi _{k, *}\mathcal{G}_ k, \quad k = 1, \ldots , n$

satisfying the following properties:

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

2. the map $\alpha _ k$ induces an isomorphism

$\kappa (\xi _ k)^{\oplus r_ k} \longrightarrow (\pi _{k, *}\mathcal{G}_ k)_{\xi _ k} \otimes _{\mathcal{O}_{Y_ k, \xi _ k}} \kappa (\xi _ k)$

where $\xi _ k \in (Y_ k)_ s$ is the unique generic point,

3. for $k = 2, \ldots , n$ the system $(Z_ k, Y_ k, i_ k, \pi _ k, \mathcal{G}_ k)$ is a one step dévissage of $\mathop{\mathrm{Coker}}(\alpha _{k - 1})/Y_{k - 1}/S$ over $s$,

4. $\mathop{\mathrm{Coker}}(\alpha _ n) = 0$.

In this case we say that $(Z_ k, Y_ k, i_ k, \pi _ k, \mathcal{G}_ k, \alpha _ k)_{k = 1, \ldots , n}$ is a complete dévissage of $\mathcal{F}/X/S$ over $s$.

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