The Stacks project

Lemma 31.11.1. Let $X$ be an integral scheme with generic point $\eta$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $U \subset X$ be nonempty open and $s \in \mathcal{F}(U)$. The following are equivalent

1. for some $x \in U$ the image of $s$ in $\mathcal{F}_ x$ is torsion,

2. for all $x \in U$ the image of $s$ in $\mathcal{F}_ x$ is torsion,

3. the image of $s$ in $\mathcal{F}_\eta$ is zero,

4. the image of $s$ in $j_*\mathcal{F}_\eta$ is zero, where $j : \eta \to X$ is the inclusion morphism.

Lemma 31.11.3. Let $X$ be an integral scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. The following are equivalent

1. $\mathcal{F}$ is torsion free,

2. for $U \subset X$ affine open $\mathcal{F}(U)$ is a torsion free $\mathcal{O}(U)$-module.

Lemma 31.11.4. Let $X$ be an integral scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. The torsion sections of $\mathcal{F}$ form a quasi-coherent $\mathcal{O}_ X$-submodule $\mathcal{F}_{tors} \subset \mathcal{F}$. The quotient module $\mathcal{F}/\mathcal{F}_{tors}$ is torsion free.

Lemma 31.11.5. Let $X$ be an integral scheme. Any flat quasi-coherent $\mathcal{O}_ X$-module is torsion free.

Lemma 31.11.6. Let $f : X \to Y$ be a flat morphism of integral schemes. Let $\mathcal{G}$ be a torsion free quasi-coherent $\mathcal{O}_ Y$-module. Then $f^*\mathcal{G}$ is a torsion free $\mathcal{O}_ X$-module.

Lemma 31.11.7. Let $f : X \to Y$ be a flat morphism of schemes. If $Y$ is integral and the generic fibre of $f$ is integral, then $X$ is integral.

Lemma 31.11.8. Let $X$ be an integral scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then $\mathcal{F}$ is torsion free if and only if $\mathcal{F}_ x$ is a torsion free $\mathcal{O}_{X, x}$-module for all $x \in X$.

Lemma 31.11.9. Let $X$ be an integral scheme. Let $0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{F}'' \to 0$ be a short exact sequence of quasi-coherent $\mathcal{O}_ X$-modules. If $\mathcal{F}$ and $\mathcal{F}''$ are torsion free, then $\mathcal{F}'$ is torsion free.

Lemma 31.11.10. Let $X$ be a locally Noetherian integral scheme with generic point $\eta$. Let $\mathcal{F}$ be a nonzero coherent $\mathcal{O}_ X$-module. The following are equivalent

1. $\mathcal{F}$ is torsion free,

2. $\eta$ is the only associated prime of $\mathcal{F}$,

3. $\eta$ is in the support of $\mathcal{F}$ and $\mathcal{F}$ has property $(S_1)$, and

4. $\eta$ is in the support of $\mathcal{F}$ and $\mathcal{F}$ has no embedded associated prime.

Lemma 31.11.11. Let $X$ be an integral regular scheme of dimension $\leq 1$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. The following are equivalent

1. $\mathcal{F}$ is torsion free,

2. $\mathcal{F}$ is finite locally free.

Lemma 31.11.12. Let $X$ be an integral scheme. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent $\mathcal{O}_ X$-modules. If $\mathcal{G}$ is torsion free and $\mathcal{F}$ is of finite presentation, then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is torsion free.

Lemma 31.11.13. Let $X$ be an integral locally Noetherian scheme. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of quasi-coherent $\mathcal{O}_ X$-modules. Assume $\mathcal{F}$ is coherent, $\mathcal{G}$ is torsion free, and that for every $x \in X$ one of the following happens

1. $\mathcal{F}_ x \to \mathcal{G}_ x$ is an isomorphism, or

2. $\text{depth}(\mathcal{F}_ x) \geq 2$.

Then $\varphi$ is an isomorphism.