Proposition 38.5.7. Let $S$ be a scheme. Let $X$ be locally of finite type over $S$. Let $x \in X$ be a point with image $s \in S$. There exists a commutative diagram

$\xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^ g \ar[d] \\ (S, s) & (S', s') \ar[l] }$

of pointed schemes such that the horizontal arrows are elementary étale neighbourhoods and such that $g^*\mathcal{F}/X'/S'$ has a complete dévissage at $x$.

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

$\xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^ g \ar[d] \\ (S, s) & (S', s') \ar[l] }$

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

$\alpha _1 : \mathcal{O}_{Y_1}^{\oplus r_1} \longrightarrow \pi _{1, *}\mathcal{G}_1$

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

38.5.7.1
$$\label{flat-equation-overcome-this} \vcenter { \xymatrix{ (Y_1, y_1) \ar[d] & (Y'_1, y'_1) \ar[l]^ h \ar[d] \\ (S, s) & (S', s') \ar[l] } }$$

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

$(Z_ k, Y_ k, i_ k, \pi _ k, \mathcal{G}_ k, \alpha _ k, z_ k, y_ k)_{k = 2, \ldots , n}$

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

$\xymatrix{ & & (X'', x'') \ar[lld] \ar[d] & (Z''_1, z''_1) \ar[l] \ar[lld] \ar[d] \\ (X, x) \ar[d] & (Z_1, z_1) \ar[l] \ar[d] & (S'', s'') \ar[lld] & (Y''_1, y''_1) \ar[lld] \ar[l] \\ (S, s) & (Y_1, y_1) \ar[l] }$

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

38.5.7.2
$$\label{flat-equation-shrink-this} (Z_ k \times _{S'} S'', Y_ k \times _{S'} S'', i''_ k, \pi ''_ k, \mathcal{G}''_ k, \alpha ''_ k, z''_ k, y''_ k)_{k = 2, \ldots , n}$$

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

$Z''_1, Y''_1, Z'_2 \times _{S'} S'', Y'_2 \times _{S'} S'', \ldots , Y'_ n \times _{S'} S''$

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

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