The Stacks project

Definition 41.9.1. Flatness of modules and rings.

  1. A module $N$ over a ring $A$ is said to be flat if the functor $M \mapsto M \otimes _ A N$ is exact.

  2. If this functor is also faithful, we say that $N$ is faithfully flat over $A$.

  3. A morphism of rings $f : A \to B$ is said to be flat (resp. faithfully flat) if the functor $M \mapsto M \otimes _ A B$ is exact (resp. faithful and exact).

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View Definition 41.9.1 as pdf