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 $n\geq 1$

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    $\begingroup$ 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. $\endgroup$ – Neil Strickland May 14 at 19:23
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    $\begingroup$ The integers are a counterexample to the second statement. $\endgroup$ – skupers May 14 at 19:37
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    $\begingroup$ Why would you ever expect those statements to be true? $\endgroup$ – Denis Nardin May 15 at 8:59
  • $\begingroup$ 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 $K$. $\endgroup$ – user127776 May 15 at 9:34
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    $\begingroup$ 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... $\endgroup$ – Denis Nardin May 17 at 7:52

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