Definition 10.39.1. Let $R$ be a ring.

An $R$-module $M$ is called

*flat*if whenever $N_1 \to N_2 \to N_3$ is an exact sequence of $R$-modules the sequence $M \otimes _ R N_1 \to M \otimes _ R N_2 \to M \otimes _ R N_3$ is exact as well.An $R$-module $M$ is called

*faithfully flat*if the complex of $R$-modules $N_1 \to N_2 \to N_3$ is exact if and only if the sequence $M \otimes _ R N_1 \to M \otimes _ R N_2 \to M \otimes _ R N_3$ is exact.A ring map $R \to S$ is called

*flat*if $S$ is flat as an $R$-module.A ring map $R \to S$ is called

*faithfully flat*if $S$ is faithfully flat as an $R$-module.

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