The Stacks project

38.10 Flat finite type modules, Part I

In some cases given a ring map $R \to S$ of finite presentation and a finite $S$-module $N$ the flatness of $N$ over $R$ implies that $N$ is of finite presentation. In this section we prove this is true “pointwise”. We remark that the first proof of Proposition 38.10.3 uses the geometric results of Section 38.3 but not the existence of a complete dévissage.

Lemma 38.10.1. Let $(R, \mathfrak m)$ be a local ring. Let $R \to S$ be a finitely presented flat ring map with geometrically integral fibres. Write $\mathfrak p = \mathfrak mS$. Let $\mathfrak q \subset S$ be a prime ideal lying over $\mathfrak m$. Let $N$ be a finite $S$-module. There exist $r \geq 0$ and an $S$-module map

\[ \alpha : S^{\oplus r} \longrightarrow N \]

such that $\alpha : \kappa (\mathfrak p)^{\oplus r} \to N \otimes _ S \kappa (\mathfrak p)$ is an isomorphism. For any such $\alpha $ the following are equivalent:

  1. $N_{\mathfrak q}$ is $R$-flat,

  2. $\alpha $ is $R$-universally injective and $\mathop{\mathrm{Coker}}(\alpha )_{\mathfrak q}$ is $R$-flat,

  3. $\alpha $ is injective and $\mathop{\mathrm{Coker}}(\alpha )_{\mathfrak q}$ is $R$-flat,

  4. $\alpha _{\mathfrak p}$ is an isomorphism and $\mathop{\mathrm{Coker}}(\alpha )_{\mathfrak q}$ is $R$-flat, and

  5. $\alpha _{\mathfrak q}$ is injective and $\mathop{\mathrm{Coker}}(\alpha )_{\mathfrak q}$ is $R$-flat.

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. $\square$

Lemma 38.10.2. Let $(R, \mathfrak m)$ be a local ring. Let $R \to S$ be a ring map of finite presentation. Let $N$ be a finite $S$-module. Let $\mathfrak q$ be a prime of $S$ lying over $\mathfrak m$. Assume that $N_{\mathfrak q}$ is flat over $R$, and assume there exists a complete dévissage of $N/S/R$ at $\mathfrak q$. Then $N$ is a finitely presented $S$-module, free as an $R$-module, and there exists an isomorphism

\[ N \cong B_1^{\oplus r_1} \oplus \ldots \oplus B_ n^{\oplus r_ n} \]

as $R$-modules where each $B_ i$ is a smooth $R$-algebra with geometrically irreducible fibres.

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. $\square$

Proposition 38.10.3. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \in X$ with image $s \in S$. Assume that

  1. $f$ is locally of finite presentation,

  2. $\mathcal{F}$ is of finite type, and

  3. $\mathcal{F}$ is flat at $x$ over $S$.

Then there exists an elementary étale neighbourhood $(S', s') \to (S, s)$ and an open subscheme

\[ V \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \]

which contains the unique point of $X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ mapping to $x$ such that the pullback of $\mathcal{F}$ to $V$ is an $\mathcal{O}_ V$-module of finite presentation and flat over $\mathcal{O}_{S', s'}$.

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

\[ \xymatrix{ X_{\mathcal{O}_{S', s'}} \ar[ddr]_ f & X'_{\mathcal{O}_{S', s'}} \ar[dd] \ar[l]^ g & Z'_{\mathcal{O}_{S', s'}} \ar[l]^ i \ar[d]^\pi \\ & & Y'_{\mathcal{O}_{S', s'}} \ar[dl]^ h \\ & \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) } \]

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 35.22.2. 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

\[ \alpha : B^{\oplus r} \to N \]

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

\[ 0 \to B^{\oplus r} \to N \to Q \to 0 \]

(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

\[ 0 \to B_ g^{\oplus r} \to N_ g \to Q_ g \to 0 \]

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

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

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 $x$. (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

\[ \Gamma (X', g^*\mathcal{F})/ \Gamma (X', \mathcal{O}_{X'})/ \Gamma (S', \mathcal{O}_{S'}) \]

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

\[ \Gamma (X', \mathcal{F}') \otimes _{\Gamma (S', \mathcal{O}_{S'})} \mathcal{O}_{S', s'} \]

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.) $\square$

Lemma 38.10.4. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. Let $s \in S$. Then the set

\[ \{ x \in X_ s \mid \mathcal{F} \text{ flat over }S\text{ at }x\} \]

is open in the fibre $X_ s$.

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 $U$. $\square$

Lemma 38.10.5. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \in X$ with image $s \in S$. Assume that

  1. $f$ is locally of finite type,

  2. $\mathcal{F}$ is of finite type, and

  3. $\mathcal{F}$ is flat at $x$ over $S$.

Then there exists an elementary étale neighbourhood $(S', s') \to (S, s)$ and an open subscheme

\[ V \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \]

which contains the unique point of $X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ mapping to $x$ such that the pullback of $\mathcal{F}$ to $V$ is flat over $\mathcal{O}_{S', s'}$.

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

\[ W \subset \mathbf{A}^ n_{\mathcal{O}_{S', s'}} \]

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. $\square$

Lemma 38.10.6. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $s \in S$. Assume that

  1. $f$ is of finite presentation,

  2. $\mathcal{F}$ is of finite type, and

  3. $\mathcal{F}$ is flat over $S$ at every point of the fibre $X_ s$.

Then there exists an elementary étale neighbourhood $(S', s') \to (S, s)$ and an open subscheme

\[ V \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \]

which contains the fibre $X_ s = X \times _ S s'$ such that the pullback of $\mathcal{F}$ to $V$ is an $\mathcal{O}_ V$-module of finite presentation and flat over $\mathcal{O}_{S', s'}$.

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 37.32.4. Set $V = \bigcup V_ i$ where $V_ i$ is the inverse images of the open $V_{x_ i}$ via the morphism

\[ X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \longrightarrow X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S_{x_ i}, s_{x_ i}}) \]

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 $\mathcal{O}_{S', s'}$. $\square$

Lemma 38.10.7. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $s \in S$. Assume that

  1. $f$ is of finite type,

  2. $\mathcal{F}$ is of finite type, and

  3. $\mathcal{F}$ is flat over $S$ at every point of the fibre $X_ s$.

Then there exists an elementary étale neighbourhood $(S', s') \to (S, s)$ and an open subscheme

\[ V \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \]

which contains the fibre $X_ s = X \times _ S s'$ such that the pullback of $\mathcal{F}$ to $V$ is flat over $\mathcal{O}_{S', s'}$.

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 37.32.4. Set $V = \bigcup V_ i$ where $V_ i$ is the inverse images of the open $V_{x_ i}$ via the morphism

\[ X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \longrightarrow X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S_{x_ i}, s_{x_ i}}) \]

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

Lemma 38.10.8. Let $S$ be a scheme. Let $X$ be locally of finite type over $S$. Let $x \in X$ with image $s \in S$. If $X$ is flat at $x$ over $S$, then there exists an elementary étale neighbourhood $(S', s') \to (S, s)$ and an open subscheme

\[ V \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \]

which contains the unique point of $X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ mapping to $x$ such that $V \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ is flat and of finite presentation.

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

\[ W \subset \mathbf{A}^ n_{\mathcal{O}_{S', s'}} \]

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. $\square$

Lemma 38.10.9. Let $f : X \to S$ be a morphism which is locally of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. If $x \in X$ and $\mathcal{F}$ is flat at $x$ over $S$, then $\mathcal{F}_ x$ is an $\mathcal{O}_{X, x}$-module of finite presentation.

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

\[ \mathcal{O}_{X, x} \longrightarrow \mathcal{O}_{X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}), x'} = \mathcal{O}_{X \times _ S S', x'} \]

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). $\square$

Lemma 38.10.10. Let $f : X \to S$ be a morphism which is locally of finite type. Let $x \in X$ with image $s \in S$. If $f$ is flat at $x$ over $S$, then $\mathcal{O}_{X, x}$ is essentially of finite presentation over $\mathcal{O}_{S, s}$.

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. $\square$


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