The Stacks project

Lemma 31.33.7. In the situation of Definition 31.33.1. Suppose that

\[ 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 \]

is an exact sequence of quasi-coherent sheaves on $X$ which remains exact after any base change $T \to S$. Then the strict transforms of $\mathcal{F}_ i'$ relative to any blowup $S' \to S$ form a short exact sequence $0 \to \mathcal{F}'_1 \to \mathcal{F}'_2 \to \mathcal{F}'_3 \to 0$ too.

Proof. We may localize on $S$ and $X$ and assume both are affine. Then we may push $\mathcal{F}_ i$ to $S$, see Lemma 31.33.4. We may assume that our blowup is the morphism $1 : S \to S$ associated to an effective Cartier divisor $D \subset S$. Then the translation into algebra is the following: Suppose that $A$ is a ring and $0 \to M_1 \to M_2 \to M_3 \to 0$ is a universally exact sequence of $A$-modules. Let $a\in A$. Then the sequence

\[ 0 \to M_1/a\text{-power torsion} \to M_2/a\text{-power torsion} \to M_3/a\text{-power torsion} \to 0 \]

is exact too. Namely, surjectivity of the last map and injectivity of the first map are immediate. The problem is exactness in the middle. Suppose that $x \in M_2$ maps to zero in $M_3/a\text{-power torsion}$. Then $y = a^ n x \in M_1$ for some $n$. Then $y$ maps to zero in $M_2/a^ nM_2$. Since $M_1 \to M_2$ is universally injective we see that $y$ maps to zero in $M_1/a^ nM_1$. Thus $y = a^ n z$ for some $z \in M_1$. Thus $a^ n(x - y) = 0$. Hence $y$ maps to the class of $x$ in $M_2/a\text{-power torsion}$ as desired. $\square$

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