- What is the definition of a flat module over a ring \(R\)?
- Give standard examples of flat modules, including an example of a flat module that is not projective/free, and an example of a non-flat module.
Solution
A right \(R\)-module \(M\) is flat if the functor \[ -\otimes_R M \] from left \(R\)-modules to abelian groups is exact. Since tensor product is always right-exact, this is equivalent to requiring that tensoring with \(M\) preserve injections. Equivalently, \[ \operatorname{Tor}_1^R(N,M)=0 \] for every left \(R\)-module \(N\).
Every free module is flat, and every projective module is flat because a projective module is a direct summand of a free module.
A standard flat nonprojective example is \(\mathbb Q\) as a \(\mathbb Z\)-module. It is a localization of \(\mathbb Z\), hence flat. It is not projective: if it were a direct summand of a free abelian group, some coordinate projection would give a nonzero homomorphism \(\mathbb Q\to\mathbb Z\), but \[ \operatorname{Hom}_{\mathbb Z}(\mathbb Q,\mathbb Z)=0. \]
A standard non-flat example is \(\mathbb Z/n\mathbb Z\) for \(n\ge2\). Tensoring the injection \[ \mathbb Z\xrightarrow{\times n}\mathbb Z \] with \(\mathbb Z/n\mathbb Z\) gives the zero map \[ \mathbb Z/n\mathbb Z\xrightarrow{0}\mathbb Z/n\mathbb Z, \] which is not injective. Thus \(\mathbb Z/n\mathbb Z\) is not flat.