Regarding the published version of "Motivic cohomology groups are isomorphic to higher Chow groups in any characterstic" (IMRN) available at:

http://imrn.oxfordjournals.org/content/2002/7/351.full.pdf

I would appreciate any elucidation on the following two points:

**(1)** a short assertion is made on the second page whose proof is not entirely clear to me

and

**(2)** how this assertion factors into the proof of Theorem 1.

**For (1)** the short assertion is (from page 2, second paragraph from top):

For any smooth scheme $X$ (of finite type) over a field $k$, $x$ a point of $X$ and $k_0$ the subfield of constants of $k$, there exists a smooth variety $X_0$ over $k_0$ and a point $x_0$ on $X_0$ such that the local rings $O_{X, x}$ and $O_{X_0, x_0}$ are isomorphic.

As noted, the dimension of $X_0$ may be larger than that of $X$. Here is an example I worked out that I think illustrates the principle of the proof:

**Example**. Let $k = \mathbb{F}_p(t), X = \mathbb{A}^1_k$ with parameter $T$, and let $x$ correspond to $(T-0)$. Then $k_0 = \mathbb{F}_p$ and put $X_0 = \mathbb{A}^2_{k_0}$ with parameters $u, v$. Then via $u\mapsto T, v\mapsto t$, and $x_0$ corresponding to $(u-0)$, we have the desired isomorphism of local rings. Note that the residue fields of the local rings are $\mathbb{F}_p(t)$ and $\mathbb{F}_p(v)$, respectively.

**In general** (*assuming $k$ is absolutely finitely generated, i.e. is finitely generated over its prime field*): since the question is local, we assume $X$ affine: $X = Spec(A)$ with $A = k[T_1, \ldots, T_n]/I$. Then take a transcendence basis $t_1, \ldots, t_r$ of $k$ over $k_0$. Then it seems that $X_0$ is built from $T_1, \ldots, T_n, t_1, \ldots, t_r$, but I haven't worked this out in general.

**Problem**: what if $k$ does not have finite transcendence degree over $k_0$? In the absolutely finitely generated case, if $F$ is the prime field of $k$, then $k_0$ is a finite extension of $F$ and $tr.deg_F(k) = tr.deg_{k_0}(k)$.

**As for (2)**, the part that is not clear to me is when one passes from $X$ to $X_0$ in the course of the proof of the above cited paper (for example, in Prop 4).

NB: the above cited assertion is *not* included in the preprint http://www.math.uiuc.edu/K-theory/378/allagree.pdf

Cf. the sentence directly after Corollary 2 on page 1 of this preprint