I asked this question on Math Stack Exchange, but received no answer. I thought that maybe I would get an answer here.

Suppose that $X$ is a smooth $n$-dimensional quasiprojective variety over $\mathbb{C}$ and that $G$ is a finite group acting on $X$. Let $Y=X/G$. Let $\text{Pic}\ Y$ denote the Picard group of $Y$ and let $A_{n-1}(Y)$ denote the Chow group of codimension 1 cycles on $Y$ mod rational equivalence. Is it true that $\text{Pic}\ Y\otimes \mathbb{Q}\cong A_{n-1}(Y)\otimes \mathbb{Q}$?

  • $\begingroup$ math.stackexchange.com/questions/1975896/… $\endgroup$
    – K.K.
    Oct 22, 2016 at 5:42
  • 4
    $\begingroup$ Yes, this is true. Since $Y$ is normal there is an injective map from $Pic(Y)$ to $A_{n-1}(Y)$. This map is surjective rationally, since if $D$ is any Weil divisor on $Y$, $cD$ is a Cartier divisor on $X$ for $c = |G|$. This can be seen by a norm argument. $\endgroup$
    – naf
    Oct 22, 2016 at 7:02


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