Lemma 10.77.6. Let $R$ be a semi-local ring. Let $M$ be a finite locally free module. If $M$ has constant rank, then $M$ is free. In particular, if $R$ has connected spectrum, then $M$ is free.
Proof. Omitted. Hints: First show that $M/\mathfrak m_ iM$ has the same dimension $d$ for all maximal ideal $\mathfrak m_1, \ldots , \mathfrak m_ n$ of $R$ using the rank is constant. Next, show that there exist elements $x_1, \ldots , x_ d \in M$ which form a basis for each $M/\mathfrak m_ iM$ by the Chinese remainder theorem. Finally show that $x_1, \ldots , x_ d$ is a basis for $M$. $\square$
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