Are fields flat $\mathbb Z$-modules?

Question

This is probably one of those questions with a super obvious counterexample, but here goes.

Is a field necessarily a flat $\mathbb Z$-module?

Multiplication with 2 is injective $\mathbb Z \rightarrow \mathbb Z$. What happens if you $\otimes_{\mathbb Z} \mathbb F_2$ — Blah, Mar 26 '12 at 20:03

Aru Ray · Accepted Answer · 2012-10-17T18:25:22.877

20

Over a PID (such as $\mathbb{Z}$) being flat is equivalent to being torsion-free. Therefore, if your field is torsion-free, it is flat, and if it has torsion, it is not flat.

edited Oct 17 '12 at 18:25

answered Mar 26 '12 at 20:01

Aru Ray

1,712

score 12 · Answer 2 · answered Mar 26 '12 at 19:59

12

A $\mathbb{Z}$-module is flat if and only if it is torsion free, so it might depend on the characteristic of the field.

answered Mar 26 '12 at 19:59

Elchanan Solomon

30,807

Are fields flat $\mathbb Z$-modules?

2 Answers2