# Chow groups of locally trivial affine fibrations

Let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic $0$. A locally trivial $\mathbf{A}^n$-fibration is a morphism $\pi \colon Y \to X$ such that $\pi^{-1}(U)\cong U\times \mathbf{A}^n$, and that $\pi\colon U\times \mathbf{A}^n \to U$ is the projection onto the second factor, for every open set $U\subseteq X$ in some Zariski open covering of $X$.

It is proved in Fulton's Intersection Theory that any such $\pi$ induces surjections on the Chow groups via pull-back $\pi^*\colon A^*(X) \to A^*(Y)$, and that this map is an isomorphism if $\pi$ admits a structure of vector bundle.

I would like to know whether $\pi^*$ is an isomorphism in general or not.

I suspect it to be so: Recall that $M^c(X)$ is the motive with compact support of $X$ in the sense of Voevodsky. He proved that there are

1. Localisation sequences for any closed subscheme $Z\subset X$ and its complement $U$ $$M^c(Z) \to M^c(X) \to M^c(U) \to,$$
2. Pull-back morphisms for any flat morphism $f\colon Y\to X$ of equidimension $n$ $$M^c(X)(n)[2n]\to M^c(Y),$$
3. Isomorphisms $M^c(X)(n)[2n]\cong M^c(X\times \mathbf{A}^n)$, and
4. Comparisons $\mathrm{Hom}_{\mathbf{DM}^-}(\mathbf{Z}(k)[2k], M^c(X))\cong A^{\dim X-k}(X)$.

It appears to me that one can apply 1., 2., and 3. to obtain an isomorphism $M^c(X)(n)[2n]\to M^c(Y)$ for the locally trivial $\mathbf{A}^n$-fibration $\pi$, and then use 4. to conclude the isomorphisms $$A^{k}(X)\cong\mathrm{Hom}_{\mathbf{DM}^-}(\mathbf{Z}(\dim X-k)[2(\dim X-k)], M^c(X))\cong\mathrm{Hom}_{\mathbf{DM}^-}(\mathbf{Z}(\dim Y-k)[2(\dim Y-k)], M^c(X)(n)[2n])\cong\mathrm{Hom}_{\mathbf{DM}^-}(\mathbf{Z}(\dim Y-k)[2(\dim Y-k)], M^c(Y))\cong A^{k}(Y).$$ Is anything wrong with the above arguments?

• Singularities?? – Konrad Voelkel May 11 '15 at 12:16
• I didn't find any assumption on the singularities imposed in each of the statements 1., 2., 3., 4. in the literature, either in Mozza-Voevodsky-Weibel, or in Voevodsky-Suslin-Friedlander. Only the resolution of singularity is assumed for the ground field. – Wille Liou May 11 '15 at 15:02
• I would prefer to write the right hand side of (4) as $A_k(X)$ (since $X$ does not have to be equi-dimensional); everything else seems to be fine. – Mikhail Bondarko May 12 '15 at 18:04
• Oh yeah, I should have written in that way. Thanks. – Wille Liou May 12 '15 at 23:07