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In May's notes on Cohen-Macaulay and Regular Local Rings, during the proof of Serre's theorem on page 9, he claims that

if $R$ is a local ring and $\phi\colon F\to F'$ is a map of finitely generated free $R$-modules, if it induces a monomorphism mod $\mathfrak{m}$ the maximal ideal of the local ring, then $\phi$ is a split homomorphism, stating that in that case $\phi$ must map a basis of $F$ into a set of $F'$ that can be extended to a basis of $F'$.

However, I haven't been able to see why this is true. Any help?

Here's the link to May's notes. http://www.math.uchicago.edu/~may/MISC/RegularLocal.pdf Thanks a lot!

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Let $F=R^m, F^{\prime} = R^n$, and consider $\phi$ as an $m\times n$-matrix. Then your assumption means that $\phi$ contains an $m\times m$-minor whose determinant does not belong to ${\mathfrak m}$. As $R$ is local, this implies that the minor is invertible over $R$, and hence you can apply base changes to $F,F^{\prime}$ to transform $\phi$ into $F\hookrightarrow F\oplus F^{\prime\prime}$.

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