Effects of the first algebraic K-theory on the higher algebraic K-theory

Is there any counterexamples known to the following statement? ($$A$$ a regular noetherian integral domain of finite Krull dimension)

If $$A^{\times}$$ is finitely generated then $$K_n(A)$$ is finitely generated for any $$n\geq 1$$

• Thomason proved that every connective spectrum is the algebraic $K$-theory of some symmetric monoidal category. Thus, if your statement is true, then it is a special feature of the symmetric monoidal categories of finitely generated projective modules over a commutative ring. My guess is that it is false, but I don't have a strong reason for that. – Neil Strickland May 14 at 19:23
• The integers are a counterexample to the second statement. – skupers May 14 at 19:37
• Why would you ever expect those statements to be true? – Denis Nardin May 15 at 8:59
• It is true for fields. I wanted to know how much of good approximation fields are for integral domains coming from coordinate ring of varieties. I think using the Quillen-gersten spectral sequence or filtration as you said might help to study how much of a good approximation $K_1$ is for higher $K$. – user127776 May 15 at 9:34
• I really doubt that $K_1(A)$ would be enough to study the finite generation of the higher K-groups. Think about how hard the Bass conjecture is, this would be reducing it to the case of $K_1$ which seems much more approachable. In general questions about the finite generation of the K-groups are really really hard... – Denis Nardin May 17 at 7:52