The Stacks project

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

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


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

  2. $j_ i : X_ i \to X$ are ├ętale 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 29.25.7). 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 29.25.13 we see that $\mathcal{F}$ is flat over $S$. $\square$

Comments (3)

Comment #4961 by awllower on

I guess in the third sentence in the first paragraph of the proof, it should be "Let be the blow-up ." Otherwise there is no mention of before that sentence.

Comment #4962 by awllower on

Sorry for the undefined control sequence. It is .

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  • 2 comment(s) on Section 38.30: Blowing up and flatness

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