Flat Module

A module over a unit ring is called flat iff the tensor product functor (or, equivalently, the tensor product functor ) is an exact functor.

For every -module, obeys the implication

which, in general, cannot be reversed.

A -module is flat iff it is torsion-free: hence and the infinite direct product are flat -modules, but they are not projective. In fact, over a Noetherian ring or a local ring, flatness implies projectivity only for finitely generated modules. This property, together with Serre's problem, allows it to be concluded that the three above implications are equivalences if is a finitely generated module over a polynomial ring , where is a field.