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

\[ 0 \to N \otimes _ R M'' \to N \otimes _ R M' \to N \otimes _ R M \to 0 \]

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

\[ \begin{matrix} & & 0
& & 0
& & 0
& & \\ & & \uparrow
& & \uparrow
& & \uparrow
& & \\ & & M''\otimes _ R N
& \to
& M' \otimes _ R N
& \to
& M \otimes _ R N
& \to
& 0
\\ & & \uparrow
& & \uparrow
& & \uparrow
& & \\ 0
& \to
& (M'')^{(I)}
& \to
& (M')^{(I)}
& \to
& M^{(I)}
& \to
& 0
\\ & & \uparrow
& & \uparrow
& & \uparrow
& & \\ & & M''\otimes _ R K
& \to
& M' \otimes _ R K
& \to
& M \otimes _ R K
& \to
& 0
\\ & & & & & & \uparrow
& & \\ & & & & & & 0
& & \end{matrix} \]

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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