Let $A$ be an algebra over some field $k$. Let $K\_P(A)$ be the Grothendieck group of the category of projective $A$-modules and $K\_F(A)$ the category of finite dimensional $A$-modules. I've been told there are examples where $K\_P(A)$ and $K\_F(A)$ have different rank, but I've never seen an example. 


Does anyone have such an example?



When are PIMS and Irreducibles not in correspondence?
Let $A=k[x]$. Then $K\_P(A)$ has rank 1 (I assume that you consider only finitely generated projective modules) and $K\_F(A)$ has infinite rank.


