The Stacks project

Proof. To obtain $\alpha $ set $r = \dim _{\kappa (\mathfrak p)} N \otimes _ S \kappa (\mathfrak p)$ and pick $x_1, \ldots , x_ r \in N$ which form a basis of $N \otimes _ S \kappa (\mathfrak p)$. Define $\alpha (s_1, \ldots , s_ r) = \sum s_ i x_ i$. This proves the existence.

Fix an $\alpha $. The most interesting implication is (1) $\Rightarrow $ (2) which we prove first. Assume (1). Because $S/\mathfrak mS$ is a domain with fraction field $\kappa (\mathfrak p)$ we see that $(S/\mathfrak mS)^{\oplus r} \to N_{\mathfrak p}/\mathfrak mN_{\mathfrak p} = N \otimes _ S \kappa (\mathfrak p)$ is injective. Hence by Lemmas 38.7.5 and 38.9.3. the map $S^{\oplus r} \to N_{\mathfrak p}$ is $R$-universally injective. It follows that $S^{\oplus r} \to N$ is $R$-universally injective, see Algebra, Lemma 10.82.10. Then also the localization $\alpha _{\mathfrak q}$ is $R$-universally injective, see Algebra, Lemma 10.82.13. We conclude that $\mathop{\mathrm{Coker}}(\alpha )_{\mathfrak q}$ is $R$-flat by Algebra, Lemma 10.82.7.

The implication (2) $\Rightarrow $ (3) is immediate. If (3) holds, then $\alpha _{\mathfrak p}$ is injective as a localization of an injective module map. By Nakayama's lemma (Algebra, Lemma 10.20.1) $\alpha _{\mathfrak p}$ is surjective too. Hence (3) $\Rightarrow $ (4). If (4) holds, then $\alpha _{\mathfrak p}$ is an isomorphism, so $\alpha $ is injective as $S_{\mathfrak q} \to S_{\mathfrak p}$ is injective. Namely, elements of $S \setminus \mathfrak p$ are nonzerodivisors on $S$ by a combination of Lemmas 38.7.6 and 38.9.3. Hence (4) $\Rightarrow $ (5). Finally, if (5) holds, then $N_{\mathfrak q}$ is $R$-flat as an extension of flat modules, see Algebra, Lemma 10.39.13. Hence (5) $\Rightarrow $ (1) and the proof is finished. $\square$

Proof. Let $(A_ i, B_ i, M_ i, \alpha _ i, \mathfrak q_ i)_{i = 1, \ldots , n}$ be the given complete dévissage. We prove the lemma by induction on $n$. Note that $N$ is finitely presented as an $S$-module if and only if $M_1$ is finitely presented as an $B_1$-module, see Remark 38.6.3. Note that $N_{\mathfrak q} \cong (M_1)_{\mathfrak q_1}$ as $R$-modules because (a) $N_{\mathfrak q} \cong (M_1)_{\mathfrak q'_1}$ where $\mathfrak q'_1$ is the unique prime in $A_1$ lying over $\mathfrak q_1$ and (b) $(A_1)_{\mathfrak q'_1} = (A_1)_{\mathfrak q_1}$ by Algebra, Lemma 10.41.11, so (c) $(M_1)_{\mathfrak q'_1} \cong (M_1)_{\mathfrak q_1}$. Hence $(M_1)_{\mathfrak q_1}$ is a flat $R$-module. Thus we may replace $(S, N)$ by $(B_1, M_1)$ in order to prove the lemma. By Lemma 38.10.1 the map $\alpha _1 : B_1^{\oplus r_1} \to M_1$ is $R$-universally injective and $\mathop{\mathrm{Coker}}(\alpha _1)_{\mathfrak q}$ is $R$-flat. Note that $(A_ i, B_ i, M_ i, \alpha _ i, \mathfrak q_ i)_{i = 2, \ldots , n}$ is a complete dévissage of $\mathop{\mathrm{Coker}}(\alpha _1)/B_1/R$ at $\mathfrak q_1$. Hence the induction hypothesis implies that $\mathop{\mathrm{Coker}}(\alpha _1)$ is finitely presented as a $B_1$-module, free as an $R$-module, and has a decomposition as in the lemma. This implies that $M_1$ is finitely presented as a $B_1$-module, see Algebra, Lemma 10.5.3. It further implies that $M_1 \cong B_1^{\oplus r_1} \oplus \mathop{\mathrm{Coker}}(\alpha _1)$ as $R$-modules, hence a decomposition as in the lemma. Finally, $B_1$ is projective as an $R$-module by Lemma 38.9.3 hence free as an $R$-module by Algebra, Theorem 10.85.4. This finishes the proof. $\square$

First proof. This proof is longer but does not use the existence of a complete dévissage. The problem is local around $x$ and $s$, hence we may assume that $X$ and $S$ are affine. During the proof we will finitely many times replace $S$ by an elementary étale neighbourhood of $(S, s)$. The goal is then to find (after such a replacement) an open $V \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ containing $x$ such that $\mathcal{F}|_ V$ is flat over $S$ and finitely presented. Of course we may also replace $S$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ at any point of the proof, i.e., we may assume $S$ is a local scheme. We will prove the proposition by induction on the integer $n = \dim _ x(\text{Supp}(\mathcal{F}_ s))$.

We can choose

1. elementary étale neighbourhoods $g : (X', x') \to (X, x)$, $e : (S', s') \to (S, s)$,

2. a commutative diagram

   \[ \xymatrix{ X \ar[dd]_ f & X' \ar[dd] \ar[l]^ g & Z' \ar[l]^ i \ar[d]^\pi \\ & & Y' \ar[d]^ h \\ S & S' \ar[l]_ e & S' \ar@{=}[l] } \]

3. a point $z' \in Z'$ with $i(z') = x'$, $y' = \pi (z')$, $h(y') = s'$,

4. a finite type quasi-coherent $\mathcal{O}_{Z'}$-module $\mathcal{G}$,

as in Lemma 38.3.2. We are going to replace $S$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$, see remarks in first paragraph of the proof. Consider the diagram

Here we have base changed the schemes $X', Z', Y'$ over $S'$ via $\mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \to S'$ and the scheme $X$ over $S$ via $\mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \to S$. It is still the case that $g$ is étale, see Lemma 38.2.2. After replacing $X$ by $X_{\mathcal{O}_{S', s'}}$, $X'$ by $X'_{\mathcal{O}_{S', s'}}$, $Z'$ by $Z'_{\mathcal{O}_{S', s'}}$, and $Y'$ by $Y'_{\mathcal{O}_{S', s'}}$ we may assume we have a diagram as Lemma 38.3.2 where in addition $S = S'$ is a local scheme with closed point $s$. By Lemmas 38.3.3 and 38.3.4 the result for $Y' \to S$, the sheaf $\pi _*\mathcal{G}$, and the point $y'$ implies the result for $X \to S$, $\mathcal{F}$ and $x$. Hence we may assume that $S$ is local and $X \to S$ is a smooth morphism of affines with geometrically irreducible fibres of dimension $n$.

The base case of the induction: $n = 0$. As $X \to S$ is smooth with geometrically irreducible fibres of dimension $0$ we see that $X \to S$ is an open immersion, see Descent, Lemma 35.25.2. As $S$ is local and the closed point is in the image of $X \to S$ we conclude that $X = S$. Thus we see that $\mathcal{F}$ corresponds to a finite flat $\mathcal{O}_{S, s}$ module. In this case the result follows from Algebra, Lemma 10.78.5 which tells us that $\mathcal{F}$ is in fact finite free.

The induction step. Assume the result holds whenever the dimension of the support in the closed fibre is $< n$. Write $S = \mathop{\mathrm{Spec}}(A)$, $X = \mathop{\mathrm{Spec}}(B)$ and $\mathcal{F} = \widetilde{N}$ for some $B$-module $N$. Note that $A$ is a local ring; denote its maximal ideal $\mathfrak m$. Then $\mathfrak p = \mathfrak mB$ is the unique minimal prime lying over $\mathfrak m$ as $X \to S$ has geometrically irreducible fibres. Finally, let $\mathfrak q \subset B$ be the prime corresponding to $x$. By Lemma 38.10.1 we can choose a map

such that $\kappa (\mathfrak p)^{\oplus r} \to N \otimes _ B \kappa (\mathfrak p)$ is an isomorphism. Moreover, as $N_{\mathfrak q}$ is $A$-flat the lemma also shows that $\alpha $ is injective and that $\mathop{\mathrm{Coker}}(\alpha )_{\mathfrak q}$ is $A$-flat. Set $Q = \mathop{\mathrm{Coker}}(\alpha )$. Note that the support of $Q/\mathfrak mQ$ does not contain $\mathfrak p$. Hence it is certainly the case that $\dim _{\mathfrak q}(\text{Supp}(Q/\mathfrak mQ)) < n$. Combining everything we know about $Q$ we see that the induction hypothesis applies to $Q$. It follows that there exists an elementary étale morphism $(S', s) \to (S, s)$ such that the conclusion holds for $Q \otimes _ A A'$ over $B \otimes _ A A'$ where $A' = \mathcal{O}_{S', s'}$. After replacing $A$ by $A'$ we have an exact sequence

(here we use that $\alpha $ is injective as mentioned above) of finite $B$-modules and we also get an element $g \in B$, $g \not\in \mathfrak q$ such that $Q_ g$ is finitely presented over $B_ g$ and flat over $A$. Since localization is exact we see that

is still exact. As $B_ g$ and $Q_ g$ are flat over $A$ we conclude that $N_ g$ is flat over $A$, see Algebra, Lemma 10.39.13, and as $B_ g$ and $Q_ g$ are finitely presented over $B_ g$ the same holds for $N_ g$, see Algebra, Lemma 10.5.3. $\square$

Second proof. We apply Proposition 38.5.7 to find a commutative diagram

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 $x$. (In particular $S'$ and $X'$ are affine.) By Morphisms, Lemma 29.25.13 we see that $g^*\mathcal{F}$ is flat at $x'$ over $S$ and by Lemma 38.2.3 we see that it is flat at $x'$ over $S'$. Via Remark 38.6.5 we deduce that

has a complete dévissage at the prime of $\Gamma (X', \mathcal{O}_{X'})$ corresponding to $x'$. We may base change this complete dévissage to the local ring $\mathcal{O}_{S', s'}$ of $\Gamma (S', \mathcal{O}_{S'})$ at the prime corresponding to $s'$. Thus Lemma 38.10.2 implies that

is flat over $\mathcal{O}_{S', s'}$ and of finite presentation over $\Gamma (X', \mathcal{O}_{X'}) \otimes _{\Gamma (S', \mathcal{O}_{S'})} \mathcal{O}_{S', s'}$. In other words, the restriction of $\mathcal{F}$ to $X' \times _{S'} \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ is of finite presentation and flat over $\mathcal{O}_{S', s'}$. Since the morphism $X' \times _{S'} \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \to X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ is étale (Lemma 38.2.2) its image $V \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ is an open subscheme, and by étale descent the restriction of $\mathcal{F}$ to $V$ is of finite presentation and flat over $\mathcal{O}_{S', s'}$. (Results used: Morphisms, Lemma 29.36.13, Descent, Lemma 35.7.3, and Morphisms, Lemma 29.25.13.) $\square$

Proof. Suppose $x \in U$. Choose an elementary étale neighbourhood $(S', s') \to (S, s)$ and open $V \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ as in Proposition 38.10.3. Note that $X_{s'} = X_ s$ as $\kappa (s) = \kappa (s')$. If $x' \in V \cap X_{s'}$, then the pullback of $\mathcal{F}$ to $X \times _ S S'$ is flat over $S'$ at $x'$. Hence $\mathcal{F}$ is flat at $x'$ over $S$, see Morphisms, Lemma 29.25.13. In other words $X_ s \cap V \subset U$ is an open neighbourhood of $x$ in $U$. $\square$

Proof. (The only difference between this and Proposition 38.10.3 is that we do not assume $f$ is of finite presentation.) The question is local on $X$ and $S$, hence we may assume $X$ and $S$ are affine. Write $X = \mathop{\mathrm{Spec}}(B)$, $S = \mathop{\mathrm{Spec}}(A)$ and write $B = A[x_1, \ldots , x_ n]/I$. In other words we obtain a closed immersion $i : X \to \mathbf{A}^ n_ S$. Denote $t = i(x) \in \mathbf{A}^ n_ S$. We may apply Proposition 38.10.3 to $\mathbf{A}^ n_ S \to S$, the sheaf $i_*\mathcal{F}$ and the point $t$. We obtain an elementary étale neighbourhood $(S', s') \to (S, s)$ and an open subscheme

such that the pullback of $i_*\mathcal{F}$ to $W$ is flat over $\mathcal{O}_{S', s'}$. This means that $V := W \cap \big (X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})\big )$ is the desired open subscheme. $\square$

Proof. For every point $x \in X_ s$ we can use Proposition 38.10.3 to find an elementary étale neighbourhood $(S_ x, s_ x) \to (S, s)$ and an open $V_ x \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S_ x, s_ x})$ such that $x \in X_ s = X \times _ S s_ x$ is contained in $V_ x$ and such that the pullback of $\mathcal{F}$ to $V_ x$ is an $\mathcal{O}_{V_ x}$-module of finite presentation and flat over $\mathcal{O}_{S_ x, s_ x}$. In particular we may view the fibre $(V_ x)_{s_ x}$ as an open neighbourhood of $x$ in $X_ s$. Because $X_ s$ is quasi-compact we can find a finite number of points $x_1, \ldots , x_ n \in X_ s$ such that $X_ s$ is the union of the $(V_{x_ i})_{s_{x_ i}}$. Choose an elementary étale neighbourhood $(S' , s') \to (S, s)$ which dominates each of the neighbourhoods $(S_{x_ i}, s_{x_ i})$, see More on Morphisms, Lemma 37.35.4. Set $V = \bigcup V_ i$ where $V_ i$ is the inverse images of the open $V_{x_ i}$ via the morphism

By construction $V$ contains $X_ s$ and by construction the pullback of $\mathcal{F}$ to $V$ is an $\mathcal{O}_ V$-module of finite presentation and flat over $\mathcal{O}_{S', s'}$. $\square$

Proof. (The only difference between this and Lemma 38.10.6 is that we do not assume $f$ is of finite presentation.) For every point $x \in X_ s$ we can use Lemma 38.10.5 to find an elementary étale neighbourhood $(S_ x, s_ x) \to (S, s)$ and an open $V_ x \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S_ x, s_ x})$ such that $x \in X_ s = X \times _ S s_ x$ is contained in $V_ x$ and such that the pullback of $\mathcal{F}$ to $V_ x$ is flat over $\mathcal{O}_{S_ x, s_ x}$. In particular we may view the fibre $(V_ x)_{s_ x}$ as an open neighbourhood of $x$ in $X_ s$. Because $X_ s$ is quasi-compact we can find a finite number of points $x_1, \ldots , x_ n \in X_ s$ such that $X_ s$ is the union of the $(V_{x_ i})_{s_{x_ i}}$. Choose an elementary étale neighbourhood $(S' , s') \to (S, s)$ which dominates each of the neighbourhoods $(S_{x_ i}, s_{x_ i})$, see More on Morphisms, Lemma 37.35.4. Set $V = \bigcup V_ i$ where $V_ i$ is the inverse images of the open $V_{x_ i}$ via the morphism

By construction $V$ contains $X_ s$ and by construction the pullback of $\mathcal{F}$ to $V$ is flat over $\mathcal{O}_{S', s'}$. $\square$

Proof. The question is local on $X$ and $S$, hence we may assume $X$ and $S$ are affine. Write $X = \mathop{\mathrm{Spec}}(B)$, $S = \mathop{\mathrm{Spec}}(A)$ and write $B = A[x_1, \ldots , x_ n]/I$. In other words we obtain a closed immersion $i : X \to \mathbf{A}^ n_ S$. Denote $t = i(x) \in \mathbf{A}^ n_ S$. We may apply Proposition 38.10.3 to $\mathbf{A}^ n_ S \to S$, the sheaf $\mathcal{F} = i_*\mathcal{O}_ X$ and the point $t$. We obtain an elementary étale neighbourhood $(S', s') \to (S, s)$ and an open subscheme

such that the pullback of $i_*\mathcal{O}_ X$ is flat and of finite presentation. This means that $V := W \cap \big (X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})\big )$ is the desired open subscheme. $\square$

Proof. Let $s = f(x)$. By Proposition 38.10.3 there exists an elementary étale neighbourhood $(S', s') \to (S, s)$ such that the pullback of $\mathcal{F}$ to $X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ is of finite presentation in a neighbourhood of the point $x' \in X_{s'} = X_ s$ corresponding to $x$. The ring map

is flat and local as a localization of an étale ring map. Hence $\mathcal{F}_ x$ is of finite presentation over $\mathcal{O}_{X, x}$ by descent, see Algebra, Lemma 10.83.2 (and also that a flat local ring map is faithfully flat, see Algebra, Lemma 10.39.17). $\square$

Proof. We may assume $X$ and $S$ affine. Write $X = \mathop{\mathrm{Spec}}(B)$, $S = \mathop{\mathrm{Spec}}(A)$ and write $B = A[x_1, \ldots , x_ n]/I$. In other words we obtain a closed immersion $i : X \to \mathbf{A}^ n_ S$. Denote $t = i(x) \in \mathbf{A}^ n_ S$. We may apply Lemma 38.10.9 to $\mathbf{A}^ n_ S \to S$, the sheaf $\mathcal{F} = i_*\mathcal{O}_ X$ and the point $t$. We conclude that $\mathcal{O}_{X, x}$ is of finite presentation over $\mathcal{O}_{\mathbf{A}^ n_ S, t}$ which implies what we want. $\square$