As I understand it, the finitistic dimension conjectue says that if a ring $A$ is finitely generated over a a commutative artinian ring $K$, then the finitistic dimension of $A$ is finite.

My question is: do we have to assume that $K$ artinian? I understand that we need to assume something on $K$, because even a commutative noetherian ring can have infinite finitistic dimension. But what if we simply require $K$ to be a commutative noetherian ring of finite Krull dimension (and hence, it has finite finitistic dimension)? Is there a known counterexample in this case?

That is, is there a ring $A$, which is finitely generated over a commutative noetherian ring $K$ of finite Krull dimension, such that $A$ has infinite finitistic dimension?

As I understand it, the finitistic dimension conjecture says that if a ring $A$ is finitely generated over a commutative artinian ring $K$, then the finitistic dimension of $A$ is finite.

My question is: do we have to assume that $K$ artinian? I understand that we need to assume something on $K$, because even a commutative noetherian ring can have infinite finitistic dimension. But what if we simply require $K$ to be a commutative noetherian ring of finite Krull dimension (and hence, it has finite finitistic dimension)? Is there a known counterexample in this case?

That is, is there a ring $A$, which is finitely generated over a commutative noetherian ring $K$ of finite Krull dimension, such that $A$ has infinite finitistic dimension?