flat R-module

Definition

An R-module $N$ is flat if the functor from the tensor product

$$-{\otimes }_{R}N:R\text{-mod}⟶R\text{-mod}$$

is exact. That is, for all short exact sequences of $R$-modules

$$0\to M_1\to M_2\to M_3\to 0$$

the sequence

$$0\to M_1{\otimes }_{R}N\to M_2{\otimes }_{R}N\to M_3{\otimes }_{R}N\to 0$$

is exact.

In other words, since by this proposition, the last three nodes are exact, flatness is a condition that for all injective $R$-module homomorphisms $M_1\hookrightarrow M_2$, the morphism

$$M_1{\otimes }_{R}N⟶M_2{\otimes }_{R}N$$

is injective.

Examples

Geometric interpretation