The Stacks project

Proof. We prove this by induction on the integer $d = \dim _ x(\text{Supp}(\mathcal{F}_ s))$. By Lemma 38.4.3 there exists a diagram

of pointed schemes such that the horizontal arrows are elementary étale neighbourhoods and such that $g^*\mathcal{F}/X'/S'$ has a one step dévissage at $x'$. The local nature of the problem implies that we may replace $(X, x) \to (S, s)$ by $(X', x') \to (S', s')$. Thus after doing so we may assume that there exists a one step dévissage $(Z_1, Y_1, i_1, \pi _1, \mathcal{G}_1)$ of $\mathcal{F}/X/S$ at $x$.

We apply Lemma 38.4.9 to find a map

which induces an isomorphism of vector spaces over $\kappa (\xi _1)$ where $\xi _1 \in Y_1$ is the unique generic point of the fibre of $Y_1$ over $s$. Moreover $\dim _{y_1}(\text{Supp}(\mathop{\mathrm{Coker}}(\alpha _1)_ s)) < d$. It may happen that the stalk of $\mathop{\mathrm{Coker}}(\alpha _1)_ s$ at $y_1$ is zero. In this case we may shrink $Y_1$ by Lemma 38.4.7 (2) and assume that $\mathop{\mathrm{Coker}}(\alpha _1) = 0$ so we obtain a complete dévissage of length zero.

Assume now that the stalk of $\mathop{\mathrm{Coker}}(\alpha _1)_ s$ at $y_1$ is not zero. In this case, by induction, there exists a commutative diagram

of pointed schemes such that the horizontal arrows are elementary étale neighbourhoods and such that $h^*\mathop{\mathrm{Coker}}(\alpha _1)/Y'_1/S'$ has a complete dévissage

at $y'_1$. (In particular $i_2 : Z_2 \to Y'_1$ is a closed immersion into $Y'_2$.) At this point we apply Lemma 38.4.8 to $S, X, \mathcal{F}, x, s$, the system $(Z_1, Y_1, i_1, \pi _1, \mathcal{G}_1)$ and diagram (38.5.7.1). We obtain a diagram

with all the properties as listed in the referenced lemma. In particular $Y''_1 \subset Y'_1 \times _{S'} S''$. Set $X_1 = Y'_1 \times _{S'} S''$ and let $\mathcal{F}_1$ denote the pullback of $\mathop{\mathrm{Coker}}(\alpha _1)$. By Lemma 38.5.4 the system

is a complete dévissage of $\mathcal{F}_1$ to $X_1$. Again, the nature of the problem allows us to replace $(X, x) \to (S, s)$ by $(X'', x'') \to (S'', s'')$. In this we see that we may assume:

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

2. there exists an $\alpha _1 : \mathcal{O}_{Y_1}^{\oplus r_1} \to \pi _{1, *}\mathcal{G}_1$ such that $\alpha \otimes \kappa (\xi _1)$ is an isomorphism,

3. $Y_1 \subset X_1$ is open, $y_1 = x_1$, and $\mathcal{F}_1|_{Y_1} \cong \mathop{\mathrm{Coker}}(\alpha _1)$, and

4. there exists a complete dévissage $(Z_ k, Y_ k, i_ k, \pi _ k, \mathcal{G}_ k, \alpha _ k, z_ k, y_ k)_{k = 2, \ldots , n}$ of $\mathcal{F}_1/X_1/S$ at $x_1$.

To finish the proof all we have to do is shrink the one step dévissage and the complete dévissage such that they fit together to a complete dévissage. (We suggest the reader do this on their own using Lemmas 38.4.7 and 38.5.6 instead of reading the proof that follows.) Since $Y_1 \subset X_1$ is an open neighbourhood of $x_1$ we may apply Lemma 38.5.6 (3) to find a standard shrinking $S', X'_1, Z'_2, Y'_2, \ldots , Y'_ n$ of the datum (d) so that $X'_1 \subset Y_1$. Note that $X'_1$ is also a standard open of the affine scheme $Y_1$. Next, we shrink the datum (a) as follows: first we shrink the base $S$ to $S'$, see Lemma 38.4.7 (1) and then we shrink the result to $S''$, $X''$, $Z''_1$, $Y''_1$ using Lemma 38.4.7 (2) such that eventually $Y''_1 = X'_1 \times _ S S''$ and $S'' \subset S'$. Then we see that

gives the complete dévissage we were looking for. $\square$