31.11 Torsion free modules
This section is the analogue of More on Algebra, Section 15.22 for quasi-coherent modules.
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
for some $x \in U$ the image of $s$ in $\mathcal{F}_ x$ is torsion,
for all $x \in U$ the image of $s$ in $\mathcal{F}_ x$ is torsion,
the image of $s$ in $\mathcal{F}_\eta $ is zero,
the image of $s$ in $j_*\mathcal{F}_\eta $ is zero, where $j : \eta \to X$ is the inclusion morphism.
Proof.
Omitted.
$\square$
Definition 31.11.2. Let $X$ be an integral scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module.
We say a local section of $\mathcal{F}$ is torsion if it satisfies the equivalent conditions of Lemma 31.11.1.
We say $\mathcal{F}$ is torsion free if every torsion section of $\mathcal{F}$ is $0$.
Here is the obligatory lemma comparing this to the usual algebraic notion.
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
$\mathcal{F}$ is torsion free,
for $U \subset X$ affine open $\mathcal{F}(U)$ is a torsion free $\mathcal{O}(U)$-module.
Proof.
Omitted.
$\square$
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.
Proof.
Omitted. See More on Algebra, Lemma 15.22.2 for the algebraic analogue.
$\square$
Lemma 31.11.5. Let $X$ be an integral scheme. Any flat quasi-coherent $\mathcal{O}_ X$-module is torsion free.
Proof.
Omitted. See More on Algebra, Lemma 15.22.9.
$\square$
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.
Proof.
Omitted. See More on Algebra, Lemma 15.23.7 for the algebraic analogue.
$\square$
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.
Proof.
The algebraic analogue is this: let $A$ be a domain with fraction field $K$ and let $B$ be a flat $A$-algebra such that $B \otimes _ A K$ is a domain. Then $B$ is a domain. This is true because $B$ is torsion free by More on Algebra, Lemma 15.22.9 and hence $B \subset B \otimes _ A K$.
$\square$
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 $x \in X$.
Proof.
Omitted. See More on Algebra, Lemma 15.22.6.
$\square$
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.
Proof.
Omitted. See More on Algebra, Lemma 15.22.5 for the algebraic analogue.
$\square$
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
$\mathcal{F}$ is torsion free,
$\eta $ is the only associated prime of $\mathcal{F}$,
$\eta $ is in the support of $\mathcal{F}$ and $\mathcal{F}$ has property $(S_1)$, and
$\eta $ is in the support of $\mathcal{F}$ and $\mathcal{F}$ has no embedded associated prime.
Proof.
This is a translation of More on Algebra, Lemma 15.22.8 into the language of schemes. We omit the translation.
$\square$
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
$\mathcal{F}$ is torsion free,
$\mathcal{F}$ is finite locally free.
Proof.
It is clear that a finite locally free module is torsion free. For the converse, we will show that if $\mathcal{F}$ is torsion free, then $\mathcal{F}_ x$ is a free $\mathcal{O}_{X, x}$-module for all $x \in X$. This is enough by Algebra, Lemma 10.78.2 and the fact that $\mathcal{F}$ is coherent. If $\dim (\mathcal{O}_{X, x}) = 0$, then $\mathcal{O}_{X, x}$ is a field and the statement is clear. If $\dim (\mathcal{O}_{X, x}) = 1$, then $\mathcal{O}_{X, x}$ is a discrete valuation ring (Algebra, Lemma 10.119.7) and $\mathcal{F}_ x$ is torsion free. Hence $\mathcal{F}_ x$ is free by More on Algebra, Lemma 15.22.11.
$\square$
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.
Proof.
The statement makes sense because $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is quasi-coherent by Schemes, Section 26.24. To see the statement is true, see More on Algebra, Lemma 15.22.12. Some details omitted.
$\square$
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
$\mathcal{F}_ x \to \mathcal{G}_ x$ is an isomorphism, or
$\text{depth}(\mathcal{F}_ x) \geq 2$.
Then $\varphi $ is an isomorphism.
Proof.
This is a translation of More on Algebra, Lemma 15.23.14 into the language of schemes.
$\square$
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