Let $A$ be a finitely generated $\mathbb Z$-algebra which is also a regular ring (https://en.wikipedia.org/wiki/Regular_ring). Consider $K_0(A)$ which has a commutative ring structure (the multiplication being induced by tensor product).
Is it true that the group of units $K_0(A)^\times$ is finitely generated as an abelian group ? If this is not true in general, then is it atleast true that $Pic(A)$ is finitely generated as an abelian group ?
(Note that if $K_0(A)^\times$ is finitely generated then so is $Pic(A)$ because it injects into $K_0(A)^\times$ )