# When are two projective modules of equal rank isomorphic?

Let $R$ be a commutative ring and let $M,N$ be two finitely generated projective $R$-modules which have equal rank (not necessarily constant). What kind of general results are there concerning the question of determining whether $M$ and $N$ are isomorphic or not? This is certainly a nontrivial question (e.g., consider the Picard group of a ring = isomorphism classes of f.g. projective modules of constant rank 1), but I'm looking for techniques for proving the (non-)existence of isomorphisms for certain modules of the same rank.

• One criteria is: If there is an epimorphism $f: M \to N$, then $M, N$ are isomorphic. Sep 15, 2011 at 18:24
• Could you give a reference for that? Sep 11, 2020 at 12:07

If $R$ is noetherian of dimension d, then we have:
The Bass Cancellation Theorem: If $M$ has rank $\ge d+1$ after localizing at any prime, and if there exists a finitely generated projective module $Q$ such that $Q\oplus M\approx Q\oplus N$, then $M\approx N$. (You probably don't even need to assume $N$ projective for this, as long as you know that $M$ is.)
Plumstead's Theorem: If $R=A[t]$ for some noetherian ring $A$, then you can replace $d+1$ with $d$.
In the non-noetherian case you can work with j-dimension instead of dimension (the j-dimension of $R$ is the length of a maximal chain of ideals, each of which is an intersection of maximal ideals).