the graded pieces of the gamma-filtration of Quillen K-theory and Chow groups of a regular scheme Let $X$ be a regular scheme and consider Grothendieck's $\gamma$-filtration $F^nK(X)$ on $K(X)$.  For the graded pieces, one has $Gr^0K(X) = CH^0(X)$ and $Gr^1K(X) = \mathrm{Pic}(X) = CH^1(X)$.  Does this continue to hold, i.e., do we have $Gr^pK(X) = CH^p(X)$?
I found that for $X/k$ smooth quasi-projective, $CH^q(X,p) \otimes \mathbf{Q} = K_p(X)^{(q)} \otimes \mathbf{Q}$, so this holds after rationalising.
 A: The map between the graded $K$-theory ring on the one hand and the Chow ring on the other is defined via Chern classes and requires denominators.  I know of no good reason to expect an integral isomorphism (or even a map), but I'm not aware of an explicit counterexample (though I'm vaguely aware that the experts think the place to look for that counterexample is over a field with large etale cohomological dimension).
On the other hand, if you replace the $\gamma$-filtration with the filtration by codimension of support, then you do get $Gr^p(X)$ as a quotient (with torsion kernel) of $Ch^p(X)=H^p(X,K_p)$ (over the integers) provided $X$ is both regular and of finite type over a field --- though it would follow from Gersten's conjecture that this holds for all regular $X$.    
A: This does not hold in general.
An explicit counterexample is given in:
Karpenko, Nikita A. Codimension 2 cycles on Severi-Brauer varieties. K-Theory 13 (1998), no. 4, 305–330. (Reviewer: Jean-Pierre Tignol) 16K20 (14C15 19E15). (Available on the author's webpage https://sites.ualberta.ca/~karpenko/publ/ch2.pdf).
Fix a field $k$. Let $p$ be an odd prime. Let $A$ be a central division $k$-algebra of degree $p^2$, exponent $p$, and assume $A$ decomposes into a product $D_1\otimes D_2$ of two smaller algebras (both necessarily of degree $p$). Let $X$ be the Severi-Brauer variety associated to $A$. By Proposition 4.7 of the article above, the quotient $\text{Gr}^2K(X)$ contains torsion but, by Proposition 5.3 of the article above, the group $\text{CH}^2(X)$ is torsion free.
Similarly, one can show if $A$ is a division algebra of index $8$ and exponent $2$ decomposing into a product of smaller algebras then the same conclusion holds. A more general statement is true for the prime $2$ but it depends on the indices of the tensor powers of $A$.
