The Stacks project

38.30 Blowing up and flatness

In this section we continue our discussion of results of the form: “After a blowup the strict transform becomes flat”, see More on Algebra, Section 15.26 and Divisors, Section 31.35. We will use the following (more or less standard) notation in this section. If $X \to S$ is a morphism of schemes, $\mathcal{F}$ is a quasi-coherent module on $X$, and $T \to S$ is a morphism of schemes, then we denote $\mathcal{F}_ T$ the pullback of $\mathcal{F}$ to the base change $X_ T = X \times _ S T$.

Remark 38.30.1. Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent module on $X$. Let $U \subset S$ be a quasi-compact open subscheme. Given a $U$-admissible blowup $S' \to S$ we denote $X'$ the strict transform of $X$ and $\mathcal{F}'$ the strict transform of $\mathcal{F}$ which we think of as a quasi-coherent module on $X'$ (via Divisors, Lemma 31.33.2). Let $P$ be a property of $\mathcal{F}/X/S$ which is stable under strict transform (as above) for $U$-admissible blowups. The general problem in this section is: Show (under auxiliary conditions on $\mathcal{F}/X/S$) there exists a $U$-admissible blowup $S' \to S$ such that the strict transform $\mathcal{F}'/X'/S'$ has $P$.

The general strategy will be to use that a composition of $U$-admissible blowups is a $U$-admissible blowup, see Divisors, Lemma 31.34.2. In fact, we will make use of the more precise Divisors, Lemma 31.32.14 and combine it with Divisors, Lemma 31.33.6. The result is that it suffices to find a sequence of $U$-admissible blowups

\[ S = S_0 \leftarrow S_1 \leftarrow \ldots \leftarrow S_ n \]

such that, setting $\mathcal{F}_0 = \mathcal{F}$ and $X_0 = X$ and setting $\mathcal{F}_ i/X_ i$ equal to the strict transform of $\mathcal{F}_{i - 1}/X_{i - 1}$, we arrive at $\mathcal{F}_ n/X_ n/S_ n$ with property $P$.

In particular, choose a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ S$ such that $V(\mathcal{I}) = S \setminus U$, see Properties, Lemma 28.24.1. Let $S' \to S$ be the blowup in $\mathcal{I}$ and let $E \subset S'$ be the exceptional divisor (Divisors, Lemma 31.32.4). Then we see that we've reduced the problem to the case where there exists an effective Cartier divisor $D \subset S$ whose support is $X \setminus U$. In particular we may assume $U$ is scheme theoretically dense in $S$ (Divisors, Lemma 31.13.4).

Suppose that $P$ is local on $S$: If $S = \bigcup S_ i$ is a finite open covering by quasi-compact opens and $P$ holds for $\mathcal{F}_{S_ i}/X_{S_ i}/S_ i$ then $P$ holds for $\mathcal{F}/X/S$. In this case the general problem above is local on $S$ as well, i.e., if given $s \in S$ we can find a quasi-compact open neighbourhood $W$ of $s$ such that the problem for $\mathcal{F}_ W/X_ W/W$ is solvable, then the problem is solvable for $\mathcal{F}/X/S$. This follows from Divisors, Lemmas 31.34.3 and 31.34.4.

Lemma 38.30.2. Let $R$ be a ring and let $f \in R$. Let $r\geq 0$ be an integer. Let $R \to S$ be a ring map and let $M$ be an $S$-module. Assume

  1. $R \to S$ is of finite presentation and flat,

  2. every fibre ring $S \otimes _ R \kappa (\mathfrak p)$ is geometrically integral over $R$,

  3. $M$ is a finite $S$-module,

  4. $M_ f$ is a finitely presented $S_ f$-module,

  5. for all $\mathfrak p \in R$, $f \not\in \mathfrak p$ with $\mathfrak q = \mathfrak pS$ the module $M_{\mathfrak q}$ is free of rank $r$ over $S_\mathfrak q$.

Then there exists a finitely generated ideal $I \subset R$ with $V(f) = V(I)$ such that for all $a \in I$ with $R' = R[\frac{I}{a}]$ the quotient

\[ M' = (M \otimes _ R R')/a\text{-power torsion} \]

over $S' = S \otimes _ R R'$ satisfies the following: for every prime $\mathfrak p' \subset R'$ there exists a $g \in S'$, $g \not\in \mathfrak p'S'$ such that $M'_ g$ is a free $S'_ g$-module of rank $r$.

Proof. This lemma is a generalization of More on Algebra, Lemma 15.26.5; we urge the reader to read that proof first. Choose a surjection $S^{\oplus n} \to M$, which is possible by (1). Choose a finite submodule $K \subset \mathop{\mathrm{Ker}}(S^{\oplus n} \to M)$ such that $S^{\oplus n}/K \to M$ becomes an isomorphism after inverting $f$. This is possible by (4). Set $M_1 = S^{\oplus n}/K$ and suppose we can prove the lemma for $M_1$. Say $I \subset R$ is the corresponding ideal. Then for $a \in I$ the map

\[ M_1' = (M_1 \otimes _ R R')/a\text{-power torsion} \longrightarrow M' = (M \otimes _ R R')/a\text{-power torsion} \]

is surjective. It is also an isomorphism after inverting $a$ in $R'$ as $R'_ a = R_ f$, see Algebra, Lemma 10.69.4. But $a$ is a nonzerodivisor on $M'_1$, whence the displayed map is an isomorphism. Thus it suffices to prove the lemma in case $M$ is a finitely presented $S$-module.

Assume $M$ is a finitely presented $S$-module satisfying (3). Then $J = \text{Fit}_ r(M) \subset S$ is a finitely generated ideal. By Lemma 38.9.3 we can write $S$ as a direct summand of a free $R$-module: $\bigoplus _{\alpha \in A} R = S \oplus C$. For any element $h \in S$ writing $h = \sum a_\alpha $ in the decomposition above, we say that the $a_\alpha $ are the coefficients of $h$. Let $I' \subset R$ be the ideal of coefficients of elements of $J$. Multiplication by an element of $S$ defines an $R$-linear map $S \to S$, hence $I'$ is generated by the coefficients of the generators of $J$, i.e., $I'$ is a finitely generated ideal. We claim that $I = fI'$ works.

We first check that $V(f) = V(I)$. The inclusion $V(f) \subset V(I)$ is clear. Conversely, if $f \not\in \mathfrak p$, then $\mathfrak q = \mathfrak p S$ is not an element of $V(J)$ by property (5) and More on Algebra, Lemma 15.8.6. Hence there is an element of $J$ which does not map to zero in $S \otimes _ R \kappa (\mathfrak p)$. Thus there exists an element of $I'$ which is not contained in $\mathfrak p$, so $\mathfrak p \not\in V(fI') = V(I)$.

Let $a \in I$ and set $R' = R[\frac{I}{a}]$. We may write $a = fa'$ for some $a' \in I'$. By Algebra, Lemmas 10.69.2 and 10.69.5 we see that $I' R' = a'R'$ and $a'$ is a nonzerodivisor in $R'$. Set $S' = S \otimes _ S R'$. Every element $g$ of $JS' = \text{Fit}_ r(M \otimes _ S S')$ can be written as $g = \sum _\alpha c_\alpha $ for some $c_\alpha \in I'R'$. Since $I'R' = a'R'$ we can write $c_\alpha = a'c'_\alpha $ for some $c'_\alpha \in R'$ and $g = (\sum c'_\alpha )a' = g' a'$ in $S'$. Moreover, there is an $g_0 \in J$ such that $a' = c_\alpha $ for some $\alpha $. For this element we have $g_0 = g'_0 a'$ in $S'$ where $g'_0$ is a unit in $S'$. Let $\mathfrak p' \subset R'$ be a prime ideal and $\mathfrak q' = \mathfrak p'S'$. By the above we see that $JS'_{\mathfrak q'}$ is the principal ideal generated by the nonzerodivisor $a'$. It follows from More on Algebra, Lemma 15.8.8 that $M'_{\mathfrak q'}$ can be generated by $r$ elements. Since $M'$ is finite, there exist $m_1, \ldots , m_ r \in M'$ and $g \in S'$, $g \not\in \mathfrak q'$ such that the corresponding map $(S')^{\oplus r} \to M'$ becomes surjective after inverting $g$.

Finally, consider the ideal $J' = \text{Fit}_{k - 1}(M')$. Note that $J'S'_ g$ is generated by the coefficients of relations between $m_1, \ldots , m_ r$ (compatibility of Fitting ideal with base change). Thus it suffices to show that $J' = 0$, see More on Algebra, Lemma 15.8.7. Since $R'_ a = R_ f$ (Algebra, Lemma 10.69.4) and $M'_ a = M_ f$ we see from (5) that $J'_ a$ maps to zero in $S_{\mathfrak q''}$ for any prime $\mathfrak q'' \subset S'$ of the form $\mathfrak q'' = \mathfrak p''S'$ where $\mathfrak p'' \subset R'_ a$. Since $S'_ a \subset \prod _{\mathfrak q''\text{ as above}} S'_{\mathfrak q''}$ (as $(S'_ a)_{\mathfrak p''} \subset S'_{\mathfrak q''}$ by Lemma 38.7.4) we see that $J'R'_ a = 0$. Since $a$ is a nonzerodivisor in $R'$ we conclude that $J' = 0$ and we win. $\square$

Lemma 38.30.3. Let $S$ be a quasi-compact and quasi-separated scheme. Let $X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent module on $X$. Let $U \subset S$ be a quasi-compact open. Assume

  1. $X \to S$ is affine, of finite presentation, flat, geometrically integral fibres,

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

  3. $\mathcal{F}_ U$ is of finite presentation,

  4. $\mathcal{F}$ is flat over $S$ at all generic points of fibres lying over points of $U$.

Then there exists a $U$-admissible blowup $S' \to S$ and an open subscheme $V \subset X_{S'}$ such that (a) the strict transform $\mathcal{F}'$ of $\mathcal{F}$ restricts to a finitely locally free $\mathcal{O}_ V$-module and (b) $V \to S'$ is surjective.

Proof. Given $\mathcal{F}/X/S$ and $U \subset S$ with hypotheses as in the lemma, denote $P$ the property “$\mathcal{F}$ is flat over $S$ at all generic points of fibres”. It is clear that $P$ is preserved under strict transform, see Divisors, Lemma 31.33.3 and Morphisms, Lemma 29.24.7. It is also clear that $P$ is local on $S$. Hence any and all observations of Remark 38.30.1 apply to the problem posed by the lemma.

Consider the function $r : U \to \mathbf{Z}_{\geq 0}$ which assigns to $u \in U$ the integer

\[ r(u) = \dim _{\kappa (\xi _ u)}(\mathcal{F}_{\xi _ u} \otimes \kappa (\xi _ u)) \]

where $\xi _ u$ is the generic point of the fibre $X_ u$. By More on Morphisms, Lemma 37.16.7 and the fact that the image of an open in $X_ S$ in $S$ is open, we see that $r(u)$ is locally constant. Accordingly $U = U_0 \amalg U_1 \amalg \ldots \amalg U_ c$ is a finite disjoint union of open and closed subschemes where $r$ is constant with value $i$ on $U_ i$. By Divisors, Lemma 31.34.5 we can find a $U$-admissible blowup to decompose $S$ into the disjoint union of two schemes, the first containing $U_0$ and the second $U_1 \cup \ldots \cup U_ c$. Repeating this $c - 1$ more times we may assume that $S$ is a disjoint union $S = S_0 \amalg S_1 \amalg \ldots \amalg S_ c$ with $U_ i \subset S_ i$. Thus we may assume the function $r$ defined above is constant, say with value $r$.

By Remark 38.30.1 we see that we may assume that we have an effective Cartier divisor $D \subset S$ whose support is $S \setminus U$. Another application of Remark 38.30.1 combined with Divisors, Lemma 31.13.2 tells us we may assume that $S = \mathop{\mathrm{Spec}}(R)$ and $D = \mathop{\mathrm{Spec}}(R/(f))$ for some nonzerodivisor $f \in R$. This case is handled by Lemma 38.30.2. $\square$

Lemma 38.30.4. Let $A \to C$ be a finite locally free ring map of rank $d$. Let $h \in C$ be an element such that $C_ h$ is étale over $A$. Let $J \subset C$ be an ideal. Set $I = \text{Fit}_0(C/J)$ where we think of $C/J$ as a finite $A$-module. Then $IC_ h = JJ'$ for some ideal $J' \subset C_ h$. If $J$ is finitely generated so are $I$ and $J'$.

Proof. We will use basic properties of Fitting ideals, see More on Algebra, Lemma 15.8.4. Then $IC$ is the Fitting ideal of $C/J \otimes _ A C$. Note that $C \to C \otimes _ A C$, $c \mapsto 1 \otimes c$ has a section (the multiplication map). By assumption $C \to C \otimes _ A C$ is étale at every prime in the image of $\mathop{\mathrm{Spec}}(C_ h)$ under this section. Hence the multiplication map $C \otimes _ A C_ h \to C_ h$ is étale in particular flat, see Algebra, Lemma 10.142.8. Hence there exists a $C_ h$-algebra such that $C \otimes _ A C_ h \cong C_ h \oplus C'$ as $C_ h$-algebras, see Algebra, Lemma 10.142.9. Thus $(C/J) \otimes _ A C_ h \cong (C_ h/J_ h) \oplus C'/I'$ as $C_ h$-modules for some ideal $I' \subset C'$. Hence $IC_ h = JJ'$ with $J' = \text{Fit}_0(C'/I')$ where we view $C'/J'$ as a $C_ h$-module. $\square$

Lemma 38.30.5. Let $A \to B$ be an étale ring map. Let $a \in A$ be a nonzerodivisor. Let $J \subset B$ be a finite type ideal with $V(J) \subset V(aB)$. For every $\mathfrak q \subset B$ there exists a finite type ideal $I \subset A$ with $V(I) \subset V(a)$ and $g \in B$, $g \not\in \mathfrak q$ such that $IB_ g = JJ'$ for some finite type ideal $J' \subset B_ g$.

Proof. We may replace $B$ by a principal localization at an element $g \in B$, $g \not\in \mathfrak q$. Thus we may assume that $B$ is standard étale, see Algebra, Proposition 10.142.16. Thus we may assume $B$ is a localization of $C = A[x]/(f)$ for some monic $f \in A[x]$ of some degree $d$. Say $B = C_ h$ for some $h \in C$. Choose elements $h_1, \ldots , h_ n \in C$ which generate $J$ over $B$. The condition $V(J) \subset V(aB)$ signifies that $a^ m = \sum b_ i h_ i$ in $B$ for some large $m$. Set $h_{n + 1} = a^ m$. As in Lemma 38.30.4 we take $I = \text{Fit}_0(C/(h_1, \ldots , h_{r + 1}))$. Since the module $C/(h_1, \ldots , h_{r + 1})$ is annihilated by $a^ m$ we see that $a^{dm} \in I$ which implies that $V(I) \subset V(a)$. $\square$

Lemma 38.30.6. Let $S$ be a quasi-compact and quasi-separated scheme. Let $X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent module on $X$. Let $U \subset S$ be a quasi-compact open. Assume there exist finitely many commutative diagrams

\[ \xymatrix{ & X_ i \ar[r]_{j_ i} \ar[d] & X \ar[d] \\ S_ i^* \ar[r] & S_ i \ar[r]^{e_ i} & S } \]

where

  1. $e_ i : S_ i \to S$ are quasi-compact étale morphisms and $S = \bigcup e_ i(S_ i)$,

  2. $j_ i : X_ i \to X$ are étale morphisms and $X = \bigcup j_ i(X_ i)$,

  3. $S^*_ i \to S_ i$ is an $e_ i^{-1}(U)$-admissible blowup such that the strict transform $\mathcal{F}_ i^*$ of $j_ i^*\mathcal{F}$ is flat over $S^*_ i$.

Then there exists a $U$-admissible blowup $S' \to S$ such that the strict transform of $\mathcal{F}$ is flat over $S'$.

Proof. We claim that the hypotheses of the lemma are preserved under $U$-admissible blowups. Namely, suppose $b : S' \to S$ is a $U$-admissible blowup in the quasi-coherent sheaf of ideals $\mathcal{I}$. Moreover, let $S'_ i \to S_ i$ be the blowup in the quasi-coherent sheaf of ideals $\mathcal{J}_ i$. Then the collection of morphisms $e'_ i : S'_ i = S_ i \times _ S S' \to S'$ and $j'_ i : X_ i' = X_ i \times _ S S' \to X \times _ S S'$ satisfy conditions (1), (2), (3) for the strict transform $\mathcal{F}'$ of $\mathcal{F}$ relative to the blowup $S' \to S$. First, observe that $S_ i'$ is the blowup of $S_ i$ in the pullback of $\mathcal{I}$, see Divisors, Lemma 31.32.3. Second, consider the blowup $S_ i^{\prime *} \to S_ i'$ of $S_ i'$ in the pullback of the ideal $\mathcal{J}_ i$. By Divisors, Lemma 31.32.12 we get a commutative diagram

\[ \xymatrix{ S_ i^{\prime *} \ar[r] \ar[rd] \ar[d] & S'_ i \ar[d] \\ S_ i^* \ar[r] & S_ i } \]

and all the morphisms in the diagram above are blowups. Hence by Divisors, Lemmas 31.33.3 and 31.33.6 we see

\begin{align*} & \text{ the strict transform of }(j'_ i)^*\mathcal{F}'\text{ under } S_ i^{\prime *} \to S_ i' \\ = & \text{ the strict transform of }j_ i^*\mathcal{F}\text{ under } S_ i^{\prime *} \to S_ i \\ = & \text{ the strict transform of }\mathcal{F}_ i'\text{ under } S_ i^{\prime *} \to S_ i' \\ = & \text{ the pullback of }\mathcal{F}_ i^*\text{ via } X_ i \times _{S_ i} S_ i^{\prime *} \to X_ i \end{align*}

which is therefore flat over $S_ i^{\prime *}$ (Morphisms, Lemma 29.24.7). Having said this, we see that all observations of Remark 38.30.1 apply to the problem of finding a $U$-admissible blowup such that the strict transform of $\mathcal{F}$ becomes flat over the base under assumptions as in the lemma. In particular, we may assume that $S \setminus U$ is the support of an effective Cartier divisor $D \subset S$. Another application of Remark 38.30.1 combined with Divisors, Lemma 31.13.2 shows we may assume that $S = \mathop{\mathrm{Spec}}(A)$ and $D = \mathop{\mathrm{Spec}}(A/(a))$ for some nonzerodivisor $a \in A$.

Pick an $i$ and $s \in S_ i$. Lemma 38.30.5 implies we can find an open neighbourhood $s \in W_ i \subset S_ i$ and a finite type quasi-coherent ideal $\mathcal{I} \subset \mathcal{O}_ S$ such that $\mathcal{I} \cdot \mathcal{O}_{W_ i} = \mathcal{J}_ i \mathcal{J}'_ i$ for some finite type quasi-coherent ideal $\mathcal{J}'_ i \subset \mathcal{O}_{W_ i}$ and such that $V(\mathcal{I}) \subset V(a) = S \setminus U$. Since $S_ i$ is quasi-compact we can replace $S_ i$ by a finite collection $W_1, \ldots , W_ n$ of these opens and assume that for each $i$ there exists a quasi-coherent sheaf of ideals $\mathcal{I}_ i \subset \mathcal{O}_ S$ such that $\mathcal{I}_ i \cdot \mathcal{O}_{S_ i} = \mathcal{J}_ i \mathcal{J}'_ i$ for some finite type quasi-coherent ideal $\mathcal{J}'_ i \subset \mathcal{O}_{S_ i}$. As in the discussion of the first paragraph of the proof, consider the blowup $S'$ of $S$ in the product $\mathcal{I}_1 \ldots \mathcal{I}_ n$ (this blowup is $U$-admissible by construction). The base change of $S' \to S$ to $S_ i$ is the blowup in

\[ \mathcal{J}_ i \cdot \mathcal{J}'_ i \mathcal{I}_1 \ldots \hat{\mathcal{I}_ i} \ldots \mathcal{I}_ n \]

which factors through the given blowup $S_ i^* \to S_ i$ (Divisors, Lemma 31.32.12). In the notation of the diagram above this means that $S_ i^{\prime *} = S_ i'$. Hence after replacing $S$ by $S'$ we arrive in the situation that $j_ i^*\mathcal{F}$ is flat over $S_ i$. Hence $j_ i^*\mathcal{F}$ is flat over $S$, see Lemma 38.2.3. By Morphisms, Lemma 29.24.13 we see that $\mathcal{F}$ is flat over $S$. $\square$

Theorem 38.30.7. Let $S$ be a quasi-compact and quasi-separated scheme. Let $X$ be a scheme over $S$. Let $\mathcal{F}$ be a quasi-coherent module on $X$. Let $U \subset S$ be a quasi-compact open. Assume

  1. $X$ is quasi-compact,

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

  3. $\mathcal{F}$ is a module of finite type,

  4. $\mathcal{F}_ U$ is of finite presentation, and

  5. $\mathcal{F}_ U$ is flat over $U$.

Then there exists a $U$-admissible blowup $S' \to S$ such that the strict transform $\mathcal{F}'$ of $\mathcal{F}$ is an $\mathcal{O}_{X \times _ S S'}$-module of finite presentation and flat over $S'$.

Proof. We first prove that we can find a $U$-admissible blowup such that the strict transform is flat. The question is étale local on the source and the target, see Lemma 38.30.6 for a precise statement. In particular, we may assume that $S = \mathop{\mathrm{Spec}}(R)$ and $X = \mathop{\mathrm{Spec}}(A)$ are affine. For $s \in S$ write $\mathcal{F}_ s = \mathcal{F}|_{X_ s}$ (pullback of $\mathcal{F}$ to the fibre). As $X \to S$ is of finite type $d = \max _{s \in S} \dim (\text{Supp}(\mathcal{F}_ s))$ is an integer. We will do induction on $d$.

Let $x \in X$ be a point of $X$ lying over $s \in S$ with $\dim _ x(\text{Supp}(\mathcal{F}_ s)) = d$. Apply Lemma 38.3.2 to get $g : X' \to X$, $e : S' \to S$, $i : Z' \to X'$, and $\pi : Z' \to Y'$. Observe that $Y' \to S'$ is a smooth morphism of affines with geometrically irreducible fibres of dimension $d$. Because the problem is étale local it suffices to prove the theorem for $g^*\mathcal{F}/X'/S'$. Because $i : Z' \to X'$ is a closed immersion of finite presentation (and since strict transform commutes with affine pushforward, see Divisors, Lemma 31.33.4) it suffices to prove the flattening result for $\mathcal{G}$. Since $\pi $ is finite (hence also affine) it suffices to prove the flattening result for $\pi _*\mathcal{G}/Y'/S'$. Thus we may assume that $X \to S$ is a smooth morphism of affines with geometrically irreducible fibres of dimension $d$.

Next, we apply a blowup as in Lemma 38.30.3. Doing so we reach the situation where there exists an open $V \subset X$ surjecting onto $S$ such that $\mathcal{F}|_ V$ is finite locally free. Let $\xi \in X$ be the generic point of $X_ s$. Let $r = \dim _{\kappa (\xi )} \mathcal{F}_\xi \otimes \kappa (\xi )$. Choose a map $\alpha : \mathcal{O}_ X^{\oplus r} \to \mathcal{F}$ which induces an isomorphism $\kappa (\xi )^{\oplus r} \to \mathcal{F}_\xi \otimes \kappa (\xi )$. Because $\mathcal{F}$ is locally free over $V$ we find an open neighbourhood $W$ of $\xi $ where $\alpha $ is an isomorphism. Shrink $S$ to an affine open neighbourhood of $s$ such that $W \to S$ is surjective. Say $\mathcal{F}$ is the quasi-coherent module associated to the $A$-module $N$. Since $\mathcal{F}$ is flat over $S$ at all generic points of fibres (in fact at all points of $W$), we see that

\[ \alpha _\mathfrak p : A_\mathfrak p^{\oplus r} \to N_\mathfrak p \]

is universally injective for all primes $\mathfrak p$ of $R$, see Lemma 38.10.1. Hence $\alpha $ is universally injective, see Algebra, Lemma 10.81.12. Set $\mathcal{H} = \mathop{\mathrm{Coker}}(\alpha )$. By Divisors, Lemma 31.33.7 we see that, given a $U$-admissible blowup $S' \to S$ the strict transforms of $\mathcal{F}'$ and $\mathcal{H}'$ fit into an exact sequence

\[ 0 \to \mathcal{O}_{X \times _ S S'}^{\oplus r} \to \mathcal{F}' \to \mathcal{H}' \to 0 \]

Hence Lemma 38.10.1 also shows that $\mathcal{F}'$ is flat at a point $x'$ if and only if $\mathcal{H}'$ is flat at that point. In particular $\mathcal{H}_ U$ is flat over $U$ and $\mathcal{H}_ U$ is a module of finite presentation. We may apply the induction hypothesis to $\mathcal{H}$ to see that there exists a $U$-admissible blowup such that the strict transform $\mathcal{H}'$ is flat as desired.

To finish the proof of the theorem we still have to show that $\mathcal{F}'$ is a module of finite presentation (after possibly another $U$-admissible blowup). This follows from Lemma 38.11.1 as we can assume $U \subset S$ is scheme theoretically dense (see third paragraph of Remark 38.30.1). This finishes the proof of the theorem. $\square$


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