The Stacks project

Lemma 38.27.4. In Situation 38.20.11. Let $s \in S$ let $d \geq 0$. Assume

  1. there exists a complete dévissage of $\mathcal{F}/X/S$ over some point $s \in S$,

  2. $X$ is of finite presentation over $S$,

  3. $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation, and

  4. $\mathcal{F}$ is flat in dimensions $\geq d + 1$ over $S$.

Then after possibly replacing $S$ by an open neighbourhood of $s$ the functor $F_ d$ ( is representable by a monomorphism $Z_ d \to S$ of finite presentation.

Proof. A preliminary remark is that $X$, $S$ are affine schemes and that it suffices to prove $F_ d$ is representable by a monomorphism of finite presentation $Z_ d \to S$ on the category of affine schemes over $S$. (Of course we do not require $Z_ d$ to be affine.) Hence throughout the proof of the lemma we work in the category of affine schemes over $S$.

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 $s$, see Definition 38.5.1. We will use induction on the length $n$ of the dévissage. Recall that $Y_ k \to S$ is smooth with geometrically irreducible fibres, see Definition 38.4.1. Let $d_ k$ be the relative dimension of $Y_ k$ over $S$. Recall that $i_{k, *}\mathcal{G}_ k = \mathop{\mathrm{Coker}}(\alpha _ k)$ and that $i_ k$ is a closed immersion. By the definitions referenced above we have $d_1 = \dim (\text{Supp}(\mathcal{F}_ s))$ and

\[ d_ k = \dim (\text{Supp}(\mathop{\mathrm{Coker}}(\alpha _{k - 1})_ s)) = \dim (\text{Supp}(\mathcal{G}_{k, s})) \]

for $k = 2, \ldots , n$. It follows that $d_1 > d_2 > \ldots > d_ n \geq 0$ because $\alpha _ k$ is an isomorphism in the generic point of $(Y_ k)_ s$.

Note that $i_1$ is a closed immersion and $\mathcal{F} = i_{1, *}\mathcal{G}_1$. Hence for any morphism of schemes $T \to S$ with $T$ affine, we have $\mathcal{F}_ T = i_{1, T, *}\mathcal{G}_{1, T}$ and $i_{1, T}$ is still a closed immersion of schemes over $T$. Thus $\mathcal{F}_ T$ is flat in dimensions $\geq d$ over $T$ if and only if $\mathcal{G}_{1, T}$ is flat in dimensions $\geq d$ over $T$. Because $\pi _1 : Z_1 \to Y_1$ is finite we see in the same manner that $\mathcal{G}_{1, T}$ is flat in dimensions $\geq d$ over $T$ if and only if $\pi _{1, T, *}\mathcal{G}_{1, T}$ is flat in dimensions $\geq d$ over $T$. The same arguments work for “flat in dimensions $\geq d + 1$” and we conclude in particular that $\pi _{1, *}\mathcal{G}_1$ is flat over $S$ in dimensions $\geq d + 1$ by our assumption on $\mathcal{F}$.

Suppose that $d_1 > d$. It follows from the discussion above that in particular $\pi _{1, *}\mathcal{G}_1$ is flat over $S$ at the generic point of $(Y_1)_ s$. By Lemma 38.12.1 we may replace $S$ by an affine neighbourhood of $s$ and assume that $\alpha _1$ is $S$-universally injective. Because $\alpha _1$ is $S$-universally injective, for any morphism $T \to S$ with $T$ affine, we have a short exact sequence

\[ 0 \to \mathcal{O}_{Y_{1, T}}^{\oplus r_1} \to \pi _{1, T, *}\mathcal{G}_{1, T} \to \mathop{\mathrm{Coker}}(\alpha _1)_ T \to 0 \]

and still the first arrow is $T$-universally injective. Hence the set of points of $(Y_1)_ T$ where $\pi _{1, T, *}\mathcal{G}_{1, T}$ is flat over $T$ is the same as the set of points of $(Y_1)_ T$ where $\mathop{\mathrm{Coker}}(\alpha _1)_ T$ is flat over $S$. In this way the question reduces to the sheaf $\mathop{\mathrm{Coker}}(\alpha _1)$ which has a complete dévissage of length $n - 1$ and we win by induction.

If $d_1 < d$ then $F_ d$ is represented by $S$ and we win.

The last case is the case $d_1 = d$. This case follows from a combination of Lemma 38.27.3 and Lemma 38.27.1. $\square$

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