# About freeness of modules over the coordinate ring of an affine variety

Let $X$ be an irreducible affine variety, $A$ be its coordinate ring, $M$ be an $A$-module. Suppose that for any maximal ideal $m$ of $A$, the localization $M_m$ is a free module of rank $n$ (finite and fixed) over $A_m$. Can we deduce that $M$ is free over $A$?

I am just a beginner in this field. I know from Atiyah & Macdonald's book that $M$ is always torsion-free. But I wonder if the stronger statement is still true in the specific case above. If not, what if replacing maximal ideals by prime ideals? Thanks!

• this module should be projective, but not necessarily free Oct 18, 2014 at 14:43
• Take a smooth projective curve of genus $g$ (say, over $\mathbb{C}$) minus one point. This is an affine variety, and the group of projective, rank 1 modules is isomorphic to $(\mathbb{R}/\mathbb{Z})^{2g}$.
– abx
Oct 18, 2014 at 16:25

Let $k$ be a field, $A=k[t^2,t^3]$, $\alpha$ a non-zero element of $k$, and $M$ the set of polynomials of the form $a_0+\alpha a_0 t+a_2 t^2 + ...+ a_nt^n$.