Lemma 10.39.13. Suppose that 0 \to M' \to M \to M'' \to 0 is a short exact sequence of R-modules. If M' and M'' are flat so is M. If M and M'' are flat so is M'.
Proof. We will use the criterion that a module N is flat if for every ideal I \subset R the map N \otimes _ R I \to N is injective, see Lemma 10.39.5. Consider an ideal I \subset R. Consider the diagram
\begin{matrix} 0
& \to
& M'
& \to
& M
& \to
& M''
& \to
& 0
\\ & & \uparrow
& & \uparrow
& & \uparrow
& & \\ & & M'\otimes _ R I
& \to
& M \otimes _ R I
& \to
& M''\otimes _ R I
& \to
& 0
\end{matrix}
with exact rows. This immediately proves the first assertion. The second follows because if M'' is flat then the lower left horizontal arrow is injective by Lemma 10.39.12. \square
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Comment #9876 by YoyoPan on
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