# The Stacks Project

## Tag 0AVQ

### 30.11. Torsion free modules

This section is the analogue of More on Algebra, Section 15.20 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 15.20.2 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 15.20.9. $\square$

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 15.21.7 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 15.20.9 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 15.20.6. $\square$

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 15.20.5 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 15.20.8 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 15.20.11. $\square$

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 15.20.12. 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 15.21.14 into the language of schemes. $\square$

The code snippet corresponding to this tag is a part of the file divisors.tex and is located in lines 1486–1713 (see updates for more information).

\section{Torsion free modules}
\label{section-torsion-free}

\noindent
This section is the analogue of
More on Algebra, Section \ref{more-algebra-section-torsion-flat}
for quasi-coherent modules.

\begin{lemma}
\label{lemma-torsion-sections}
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
\begin{enumerate}
\item for some $x \in U$ the image of $s$ in $\mathcal{F}_x$ is torsion,
\item for all $x \in U$ the image of $s$ in $\mathcal{F}_x$ is torsion,
\item the image of $s$ in $\mathcal{F}_\eta$ is zero,
\item the image of $s$ in $j_*\mathcal{F}_\eta$ is zero, where $j : \eta \to X$
is the inclusion morphism.
\end{enumerate}
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

\begin{definition}
\label{definition-torsion}
Let $X$ be an integral scheme. Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_X$-module.
\begin{enumerate}
\item We say a local section of $\mathcal{F}$ is {\it torsion}
if it satisfies the equivalent conditions of Lemma \ref{lemma-torsion-sections}.
\item We say $\mathcal{F}$ is {\it torsion free} if every torsion section
of $\mathcal{F}$ is $0$.
\end{enumerate}
\end{definition}

\noindent
Here is the obligatory lemma comparing this to the usual algebraic notion.

\begin{lemma}
\label{lemma-check-torsion-on-affines}
Let $X$ be an integral scheme. Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_X$-module. The following are equivalent
\begin{enumerate}
\item $\mathcal{F}$ is torsion free,
\item for $U \subset X$ affine open $\mathcal{F}(U)$
is a torsion free $\mathcal{O}(U)$-module.
\end{enumerate}
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

\begin{lemma}
\label{lemma-torsion}
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.
\end{lemma}

\begin{proof}
Omitted. See More on Algebra, Lemma \ref{more-algebra-lemma-torsion}
for the algebraic analogue.
\end{proof}

\begin{lemma}
\label{lemma-flat-torsion-free}
Let $X$ be an integral scheme. Any flat quasi-coherent $\mathcal{O}_X$-module
is torsion free.
\end{lemma}

\begin{proof}
Omitted. See More on Algebra, Lemma \ref{more-algebra-lemma-flat-torsion-free}.
\end{proof}

\begin{lemma}
\label{lemma-flat-pullback-torsion}
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.
\end{lemma}

\begin{proof}
Omitted. See
More on Algebra, Lemma \ref{more-algebra-lemma-flat-pullback-reflexive}
for the algebraic analogue.
\end{proof}

\begin{lemma}
\label{lemma-flat-over-integral-integral-fibre}
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.
\end{lemma}

\begin{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
\ref{more-algebra-lemma-flat-torsion-free}
and hence $B \subset B \otimes_A K$.
\end{proof}

\begin{lemma}
\label{lemma-check-torsion}
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 $x \in X$.
\end{lemma}

\begin{proof}
Omitted. See More on Algebra, Lemma
\ref{more-algebra-lemma-check-torsion}.
\end{proof}

\begin{lemma}
\label{lemma-extension-torsion-free}
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.
\end{lemma}

\begin{proof}
Omitted. See
More on Algebra, Lemma \ref{more-algebra-lemma-extension-torsion-free}
for the algebraic analogue.
\end{proof}

\begin{lemma}
\label{lemma-torsion-free-finite-noetherian-domain}
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
\begin{enumerate}
\item $\mathcal{F}$ is torsion free,
\item $\eta$ is the only associated prime of $\mathcal{F}$,
\item $\eta$ is in the support of $\mathcal{F}$ and $\mathcal{F}$
has property $(S_1)$, and
\item $\eta$ is in the support of $\mathcal{F}$ and $\mathcal{F}$
has no embedded associated prime.
\end{enumerate}
\end{lemma}

\begin{proof}
This is a translation of More on Algebra, Lemma
\ref{more-algebra-lemma-torsion-free-finite-noetherian-domain}
into the language of schemes. We omit the translation.
\end{proof}

\begin{lemma}
\label{lemma-torsion-free-over-regular-dim-1}
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
\begin{enumerate}
\item $\mathcal{F}$ is torsion free,
\item $\mathcal{F}$ is finite locally free.
\end{enumerate}
\end{lemma}

\begin{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 \ref{algebra-lemma-finite-projective}
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 \ref{algebra-lemma-characterize-dvr})
and $\mathcal{F}_x$ is torsion free.
Hence $\mathcal{F}_x$ is free by More on Algebra, Lemma
\ref{more-algebra-lemma-dedekind-torsion-free-flat}.
\end{proof}

\begin{lemma}
\label{lemma-hom-into-torsion-free}
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 $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is torsion free.
\end{lemma}

\begin{proof}
The statement makes sense because
$\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$
is quasi-coherent by Schemes, Section \ref{schemes-section-quasi-coherent}.
To see the statement is true, see
More on Algebra, Lemma \ref{more-algebra-lemma-hom-into-torsion-free}.
Some details omitted.
\end{proof}

\begin{lemma}
\label{lemma-isom-depth-2-torsion-free}
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
\begin{enumerate}
\item $\mathcal{F}_x \to \mathcal{G}_x$ is an isomorphism, or
\item $\text{depth}(\mathcal{F}_x) \geq 2$.
\end{enumerate}
Then $\varphi$ is an isomorphism.
\end{lemma}

\begin{proof}
This is a translation of More on Algebra, Lemma
\ref{more-algebra-lemma-isom-depth-2-torsion-free}
into the language of schemes.
\end{proof}

There are no comments yet for this tag.

## Add a comment on tag 0AVQ

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).