The Stacks project

38.5 Complete dévissage

In this section we explain what is a complete dévissage of a module and prove that such exist. The material in this section is mainly bookkeeping.

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 $s$.

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 $y_{k - 1}$.

Again we remark that a complete dévissage can only exist if $X$ and $S$ are affine.

Lemma 38.5.3. Let $S$, $X$, $\mathcal{F}$, $s$ be as in Definition 38.5.1. Let $(S', s') \to (S, s)$ be any morphism of pointed schemes. Let $(Z_ k, Y_ k, i_ k, \pi _ k, \mathcal{G}_ k, \alpha _ k)_{k = 1, \ldots , n}$ be a complete dévissage of $\mathcal{F}/X/S$ over $s$. Given this data let $X', Z'_ k, Y'_ k, i'_ k, \pi '_ k$ be the base changes of $X, Z_ k, Y_ k, i_ k, \pi _ k$ via $S' \to S$. Let $\mathcal{F}'$ be the pullback of $\mathcal{F}$ to $X'$ and let $\mathcal{G}'_ k$ be the pullback of $\mathcal{G}_ k$ to $Z'_ k$. Let $\alpha '_ k$ be the pullback of $\alpha _ k$ to $Y'_ k$. If $S'$ is affine, then $(Z'_ k, Y'_ k, i'_ k, \pi '_ k, \mathcal{G}'_ k, \alpha '_ k)_{k = 1, \ldots , n}$ is a complete dévissage of $\mathcal{F}'/X'/S'$ over $s'$.

Proof. By Lemma 38.4.4 we know that the base change of a one step dévissage is a one step dévissage. Hence it suffices to prove that formation of $\mathop{\mathrm{Coker}}(\alpha _ k)$ commutes with base change and that condition (2) of Definition 38.5.1 is preserved by base change. The first is true as $\pi '_{k, *}\mathcal{G}'_ k$ is the pullback of $\pi _{k, *}\mathcal{G}_ k$ (by Cohomology of Schemes, Lemma 30.5.1) and because $\otimes $ is right exact. The second because by the same token we have

\[ (\pi _{k, *}\mathcal{G}_ k)_{\xi _ k} \otimes _{\mathcal{O}_{Y_ k, \xi _ k}} \kappa (\xi _ k) \otimes _{\kappa (\xi _ k)} \kappa (\xi '_ k) \cong (\pi '_{k, *}\mathcal{G}'_ k)_{\xi '_ k} \otimes _{\mathcal{O}_{Y'_ k, \xi '_ k}} \kappa (\xi '_ k) \]

with obvious notation. $\square$

Lemma 38.5.4. Let $S$, $X$, $\mathcal{F}$, $x$, $s$ be as in Definition 38.5.2. Let $(S', s') \to (S, s)$ be a morphism of pointed schemes which induces an isomorphism $\kappa (s) = \kappa (s')$. Let $(Z_ k, Y_ k, i_ k, \pi _ k, \mathcal{G}_ k, \alpha _ k, z_ k, y_ k)_{k = 1, \ldots , n}$ be a complete dévissage of $\mathcal{F}/X/S$ at $x$. Let $(Z'_ k, Y'_ k, i'_ k, \pi '_ k, \mathcal{G}'_ k, \alpha '_ k)_{k = 1, \ldots , n}$ be as constructed in Lemma 38.5.3 and let $x' \in X'$ (resp. $z'_ k \in Z'$, $y'_ k \in Y'$) be the unique point mapping to both $x \in X$ (resp. $z_ k \in Z_ k$, $y_ k \in Y_ k$) and $s' \in S'$. If $S'$ is affine, then $(Z'_ k, Y'_ k, i'_ k, \pi '_ k, \mathcal{G}'_ k, \alpha '_ k, z'_ k, y'_ k)_{k = 1, \ldots , n}$ is a complete dévissage of $\mathcal{F}'/X'/S'$ at $x'$.

Definition 38.5.5. Let $S$, $X$, $\mathcal{F}$, $x$, $s$ be as in Definition 38.5.2. Consider a complete dévissage $(Z_ k, Y_ k, i_ k, \pi _ k, \mathcal{G}_ k, \alpha _ k, z_ k, y_ k)_{k = 1, \ldots , n}$ of $\mathcal{F}/X/S$ at $x$. Let us define a standard shrinking of this situation to be given by standard opens $S' \subset S$, $X' \subset X$, $Z'_ k \subset Z_ k$, and $Y'_ k \subset Y_ k$ such that $s_ k \in S'$, $x_ k \in X'$, $z_ k \in Z'$, and $y_ k \in Y'$ and such that

\[ (Z'_ k, Y'_ k, i'_ k, \pi '_ k, \mathcal{G}'_ k, \alpha '_ k, z_ k, y_ k)_{k = 1, \ldots , n} \]

is a one step dévissage of $\mathcal{F}'/X'/S'$ at $x$ where $\mathcal{G}'_ k = \mathcal{G}_ k|_{Z'_ k}$ and $\mathcal{F}' = \mathcal{F}|_{X'}$.

Lemma 38.5.6. With assumption and notation as in Definition 38.5.5 we have:

  1. If $S' \subset S$ is a standard open neighbourhood of $s$, then setting $X' = X_{S'}$, $Z'_ k = Z_{S'}$ and $Y'_ k = Y_{S'}$ we obtain a standard shrinking.

  2. Let $W \subset Y_ n$ be a standard open neighbourhood of $y$. Then there exists a standard shrinking with $Y'_ n = W \times _ S S'$.

  3. Let $U \subset X$ be an open neighbourhood of $x$. Then there exists a standard shrinking with $X' \subset U$.

Proof. Part (1) is immediate from Lemmas 38.5.4 and 38.4.7.

Proof of (2). For convenience denote $X = Y_0$. We apply Lemma 38.4.7 (2) to find a standard shrinking $S', Y'_{n - 1}, Z'_ n, Y'_ n$ of the one step dévissage of $\mathop{\mathrm{Coker}}(\alpha _{n - 1})/Y_{n - 1}/S$ at $y_{n - 1}$ with $Y'_ n = W \times _ S S'$. We may repeat this procedure and find a standard shrinking $S'', Y''_{n - 2}, Z''_{n - 1}, Y''_{n - 1}$ of the one step dévissage of $\mathop{\mathrm{Coker}}(\alpha _{n - 2})/Y_{n - 2}/S$ at $y_{n - 2}$ with $Y''_{n - 1} = Y'_{n - 1} \times _ S S''$. We may continue in this manner until we obtain $S^{(n)}, Y^{(n)}_0, Z^{(n)}_1, Y^{(n)}_1$. At this point it is clear that we obtain our desired standard shrinking by taking $S^{(n)}$, $X^{(n)}$, $Z_ k^{(n - k)} \times _ S S^{(n)}$, and $Y_ k^{(n - k)} \times _ S S^{(n)}$ with the desired property.

Proof of (3). We use induction on the length of the complete dévissage. First we apply Lemma 38.4.7 (3) to find a standard shrinking $S', X', Z'_1, Y'_1$ of the one step dévissage of $\mathcal{F}/X/S$ at $x$ with $X' \subset U$. If $n = 1$, then we are done. If $n > 1$, then by induction we can find a standard shrinking $S''$, $Y''_1$, $Z''_ k$, and $Y''_ k$ of the complete dévissage $(Z_ k, Y_ k, i_ k, \pi _ k, \mathcal{G}_ k, \alpha _ k, z_ k, y_ k)_{k = 2, \ldots , n}$ of $\mathop{\mathrm{Coker}}(\alpha _1)/Y_1/S$ at $x$ such that $Y''_1 \subset Y'_1$. Using Lemma 38.4.7 (2) we can find $S''' \subset S'$, $X''' \subset X'$, $Z'''_1$ and $Y'''_1 = Y''_1 \times _ S S'''$ which is a standard shrinking. The solution to our problem is to take

\[ S''', X''', Z'''_1, Y'''_1, Z''_2 \times _ S S''', Y''_2 \times _ S S''', \ldots , Z''_ n \times _ S S''', Y''_ n \times _ S S''' \]

This ends the proof of the lemma. $\square$

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
\begin{equation} \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] } } \end{equation}

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
\begin{equation} \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} \end{equation}

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. $\square$

Some more bookkeeping gives the following consequence.

Lemma 38.5.8. Let $X \to S$ be a finite type morphism of schemes. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $s \in S$ be a point. There exists an elementary étale neighbourhood $(S', s') \to (S, s)$ and étale morphisms $h_ i : Y_ i \to X_{S'}$, $i = 1, \ldots , n$ such that for each $i$ there exists a complete dévissage of $\mathcal{F}_ i/Y_ i/S'$ over $s'$, where $\mathcal{F}_ i$ is the pullback of $\mathcal{F}$ to $Y_ i$ and such that $X_ s = (X_{S'})_{s'} \subset \bigcup h_ i(Y_ i)$.

Proof. For every point $x \in X_ s$ we can find 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 complete dévissage at $x'$. As $X \to S$ is of finite type the fibre $X_ s$ is quasi-compact, and since each $g : X' \to X$ as above is open we can cover $X_ s$ by a finite union of $g(X'_{s'})$. Thus we can find a finite family of such diagrams

\[ \vcenter { \xymatrix{ (X, x) \ar[d] & (X'_ i, x'_ i) \ar[l]^{g_ i} \ar[d] \\ (S, s) & (S'_ i, s'_ i) \ar[l] } } \quad i = 1, \ldots , n \]

such that $X_ s = \bigcup g_ i(X'_ i)$. Set $S' = S'_1 \times _ S \ldots \times _ S S'_ n$ and let $Y_ i = X_ i \times _{S'_ i} S'$ be the base change of $X'_ i$ to $S'$. By Lemma 38.5.3 we see that the pullback of $\mathcal{F}$ to $Y_ i$ has a complete dévissage over $s$ and we win. $\square$

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