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30.11 Torsion free modules

This section is the analogue of More on Algebra, Section 15.22 for quasi-coherent modules.

Lemma 30.11.1. 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.

Proof. Omitted. $\square$

Definition 30.11.2. Let $X$ be an integral scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module.

  1. We say a local section of $\mathcal{F}$ is torsion if it satisfies the equivalent conditions of Lemma 30.11.1.

  2. 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 30.11.3. 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.

Proof. Omitted. $\square$

Lemma 30.11.4. 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 30.11.5. 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 30.11.6. 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 30.11.7. 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 30.11.8. 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 30.11.9. 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 30.11.10. 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.

Proof. This is a translation of More on Algebra, Lemma 15.22.8 into the language of schemes. We omit the translation. $\square$

Lemma 30.11.11. 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.

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 10.77.2 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 10.118.7) and $\mathcal{F}_ x$ is torsion free. Hence $\mathcal{F}_ x$ is free by More on Algebra, Lemma 15.22.11. $\square$

Lemma 30.11.12. 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 25.24. To see the statement is true, see More on Algebra, Lemma 15.22.12. Some details omitted. $\square$

Lemma 30.11.13. 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.

Proof. This is a translation of More on Algebra, Lemma 15.23.14 into the language of schemes. $\square$


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