Ring theoretic properties of the rings $K_0$.

Question

Let $A$ be a commutative ring, and $K_0(A)$ its algebraic K-theory ring. Are there are any notable results asserting ring-theoretic properties (being Noetherian, reduced, Krull dimension, etc..) of $K_0(A)$ for a (certain) class of rings $A$? The rings $K_0(A)$ are a special kind of commutative rings (e.g., they are $\lambda$-rings), so one can expect some interesting results.

Answer 1

Let me write this out as a partial answer. I will work with only $A$ which are co-ordinate rings of an affine open subset of a smooth projective variety $X$ over $\mathbb{C}$. Then, as I said, $K_0(A)$ modulo nilpotents is $\mathbb{Z}$ and thus always have Krull dimension one. It is Noetherian if and only if it is a finitely generated module over $\mathbb{Z}$. Further, if it is Noetherian then $H^i(\mathcal{O}_X)=0$ for $i>0$.