The Chow-group description of the projective generators Ck(X)

Let $X$ be a smooth proper $k$-variety, and let $C_{k(X)}$ denote the projective-generator sheaf associated with the function field $k(X)$. For a smooth $k$-variety $Y$, write $X_{k(Y)}$ for the base change of $X$ to $k(Y)$ and $CH_0(X_{k(Y)})_{\mathbb Q}$ for its zero-cycle Chow group with rational coefficients. Chow-group description. The sheaf $C_{k(X)}$ coincides with the sheaf $Y\mapsto CH_0(X_{k(Y)})_{\mathbb Q}$. This identifies the projective generators of the category with rational zero-cycle functors, but the source gives no resolution status beyond stating the claim.