Lemma 10.39.10. Let R \to S be a ring map. Let M be an S-module. If M is flat as an R-module and faithfully flat as an S-module, then R \to S is flat.
Proof. Let N_1 \to N_2 \to N_3 be an exact sequence of R-modules. By assumption N_1 \otimes _ R M \to N_2 \otimes _ R M \to N_3 \otimes _ R M is exact. We may write this as
N_1 \otimes _ R S \otimes _ S M \to N_2 \otimes _ R S \otimes _ S M \to N_3 \otimes _ R S \otimes _ S M.
By faithful flatness of M over S we conclude that N_1 \otimes _ R S \to N_2 \otimes _ R S \to N_3 \otimes _ R S is exact. Hence R \to S is flat. \square
Comments (0)
There are also: