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?