Lemma 15.22.11. Let $A$ be a Dedekind domain (for example a discrete valuation ring or more generally a PID).

1. An $A$-module is flat if and only if it is torsion free.

2. A finite torsion free $A$-module is finite locally free.

3. A finite torsion free $A$-module is finite free if $A$ is a PID.

Proof. (For the parenthetical remark in the statement of the lemma, see Algebra, Lemma 10.120.15.) Proof of (1). By Lemma 15.22.6 and Algebra, Lemma 10.39.18 it suffices to check the statement over $A_\mathfrak m$ for $\mathfrak m \subset A$ maximal. Since $A_\mathfrak m$ is a discrete valuation ring (Algebra, Lemma 10.120.17) we win by Lemma 15.22.10.

Proof of (2). Follows from Algebra, Lemma 10.78.2 and (1).

Proof of (3). Let $A$ be a PID and let $M$ be a finite torsion free module. By Lemma 15.22.7 we see that $M \subset A^{\oplus n}$ for some $n$. We argue that $M$ is free by induction on $n$. The case $n = 1$ expresses exactly the fact that $A$ is a PID. If $n > 1$ let $M' \subset R^{\oplus n - 1}$ be the image of the projection onto the last $n - 1$ summands of $R^{\oplus n}$. Then we obtain a short exact sequence $0 \to I \to M \to M' \to 0$ where $I$ is the intersection of $M$ with the first summand $R$ of $R^{\oplus n}$. By induction we see that $M$ is an extension of finite free $R$-modules, whence finite free. $\square$

Comment #3540 by Laurent Moret-Bailly on

Typo in proof of (3): "induction on $M$". Also, why write "PID or discrete valuation ring" since one class contains the other?

Comment #6769 by Oliver Roendigs on

The first sentence after "Proof of (1)." indicates that a previous version of the Lemma addressed only PIDs. It can be removed.

Comment #6931 by on

OK, I hope you are going to laugh at my explanation: when a lemma of the stacks project says "I X has B (for example if X has A), then..." in the statement of a lemma, then we try in the proof to explain why A $\Rightarrow$ B. So the proof of the lemma needs to explain the parenthetical remark, but it isn't part of the proof itself. So what I have done is move the reference to Lemma 10.120.15 to a parenthetical remark in the proof itself. See this commit.

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