Projective modules over semi-local rings

Question

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

Answer 1

If R is commutative(even not Noetherian), I think the answer is yes. Please see the paper of Hinohara, Projective modules over semilocal rings.

Answer 2

This follows both from Forster's Theorem and from Serre's Splitting Off Theorem.

If $R$ is a commutative ring, Noetherian or not, and $M$ is an $R$-module, let $\mu(M,R)$ be its minimal number of generators ($0\leq \mu(M,R)\leq\infty$).

A version of Forster's Theorem states that $\mu(M,R) \leq \max(\mu(M_{\mathfrak m},R_{\mathfrak m}))+\dim(R/\mathrm{rad}(R))$ if $M$ has finite presentation. The maximum is taken over the maximal ideals $\mathfrak m$ of $R$, and $\dim$ is Krull dimension.

If $M$ is finitely generated and projective, then $\mu(M_{\mathfrak m},R_{\mathfrak m})$ is just the $\mathfrak m$-rank of $M$, and $M$ admits finite presention. And if $R$ is semi-local, we have $\dim(R/\mathrm{rad}(R)) = 0$. It follows that when $n$ is the largest of the $\mu(M_{\mathfrak m},R_{\mathfrak m})$, there is a projection $R^{n}\twoheadrightarrow M$, giving $R^{n}=M\oplus P$ by projectivity. And when all the $\mu(M_{\mathfrak m},R_{\mathfrak m})$ are equal to $n$, $P$ must be the zero module, and $M$ is free.

Serre's Splitting Off Theorem says that if $m:=\min(\mu(M_{\mathfrak m},R_{\mathfrak m}))- \dim(R/\mathrm{rad}(R))>0$ for a finitely generated projective module $M$, then $M=R^{m}\oplus P$ with $\min(\mu(P_{\mathfrak m},R_{\mathfrak m}))=\dim(R/\mathrm{rad}(R))$, for a suitable submodule $P$. So $\mu(M_{\mathfrak m},R_{\mathfrak m})=m+\mu(P_{\mathfrak m},R_{\mathfrak m})$, and if $R$ is semi-local, $\mu(P_{\mathfrak m},R_{\mathfrak m})=0$ for some maximal ideal $\mathfrak m$. But if the $\mu(M_{\mathfrak m},R_{\mathfrak m})$ are all equal, so are the $\mu(P_{\mathfrak m},R_{\mathfrak m})$, and necessarily $P=0$.

These observations are the generalizations to semi-local rings of the well-known corresponding results for local rings.

The proofs of the above theorems are quite elementary, based on the elementary characterization of Krull dimension given by Coquand, Lombardi, Quitté et alii. Cf. http://hlombardi.free.fr/publis/KroBasSer.pdf, where these results are shown in slightly stronger form using $Hdim(R)$, the Heitmann dimension, a number less than or equal to $\dim(R/\mathrm{rad}(R))$.