Definition 10.39.1. Let $R$ be a ring.

1. 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.

2. 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.

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

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

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