Lemma 10.39.12. Suppose that R is a ring, 0 \to M'' \to M' \to M \to 0 a short exact sequence, and N an R-module. If M is flat then N \otimes _ R M'' \to N \otimes _ R M' is injective, i.e., the sequence
is a short exact sequence.
Lemma 10.39.12. Suppose that R is a ring, 0 \to M'' \to M' \to M \to 0 a short exact sequence, and N an R-module. If M is flat then N \otimes _ R M'' \to N \otimes _ R M' is injective, i.e., the sequence
is a short exact sequence.
Proof. Let R^{(I)} \to N be a surjection from a free module onto N with kernel K. The result follows from the snake lemma applied to the following diagram
with exact rows and columns. The middle row is exact because tensoring with the free module R^{(I)} is exact. \square
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