The Stacks project

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.

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