TITLE: Projective modules over a semi-local ring
QUESTION [6 upvotes]: I need a little bit of help, I found that theorem, but the book doesn't prove it and gives a reference to another book that I don't have; does anyone have an idea?

Let $R$ be a semi-local ring, and $M$ a finite projective $R$-module. Show that $M$ is free if the localizations $M_m$ have the same rank for all maximal ideals $m$ of $R$.

REPLY [3 votes]: See Lemma 1.4.4 in Bruns and Herzog, Cohen-Macaulay Rings.
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