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I work in the category of varieties over some field of characteristic zero. Assume that for any variety I can define the group $\widetilde{CH}^r(X,n)$ which behave like classical Bloch's higher Chow group. More prisiesly it satisfy the following properties:

  1. For any $X$, there is the natural map $\psi_X\colon {CH}^r(X,n)\to \widetilde{CH}^r(X,n)$
  2. When $n=2r$ the map $\psi_X$ is an isomorphism

Moreover the group $\widetilde{CH}^r(X,n)$:

  1. Contravariantly fucntorial for flat morphism and covariantly functorial for proper morphism
  2. For any embedding of closed subset, there is the localisation sequence
  3. We have $\widetilde{CH}^r(X\times\mathbb A^1,n)\cong \widetilde{CH}^r(X,n)$

Can I deduce that $\psi_X$ is an isomorphism for any $X$? Or what additional property I need to check?

I think this would be not difficult to show if I knew that $\psi_X$ is an isomorphism when $X$ is prime spectrum of some field. But I don't know that.

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  • $\begingroup$ I guess you’ve got the functoriality in 3. back-to-front. $\endgroup$ Oct 21, 2023 at 14:52
  • $\begingroup$ Yes, thank you. $\endgroup$ Oct 21, 2023 at 14:56

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