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Are there any examples known of an affine variety $A$ over an algebraically closed field such some Chow group (say, of codimension at least $2$) of $A$ with coefficients in $\mathbb{Z}/n\mathbb{Z}$ ($n$ is prime to the residue field characteristic) is non-zero?

flag	asked Sep 27 2011 at 8:43 Mikhail Bondarko 5,876●1●4●21

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By a result of Jouanaolou ("Jouanolou's trick", see his article in LNM 341), every quasi-projective variety $X$ over a field (say) has a vector bundle $V_0$ and a torsor $V$ under $V_0$, such that $V$ is an affine scheme. If $X$ is regular, then the pull-back from $X$ to $V$ will induce an isomorphism ${\rm CH}(X)_{\bf Q}\simeq {\rm CH}(V)_{\bf Q}$, so this construction should provide you with a lot of examples. – Damian Rössler Sep 27 2011 at 10:28

@Damian: Fantastic! – Wanderer Sep 27 2011 at 10:49

Great! So, I can start with a projective space.:) – Mikhail Bondarko Oct 4 2011 at 14:47

ulrich · Accepted Answer · 2011-09-27 09:16:19Z UTC

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I think there will be many such varieties.

For example, let $Q$ be a smooth $4$-dimensional projective quadric, $Q'$ a hyperplane section of $Q$ and $A = Q \backslash Q'$. For any $i$, we have the localisation sequence

$$ CH^{i-1}(Q') \to CH^i(Q) \to CH^i(A) \to 0 \ . $$

Since $CH^1(Q') = \mathbb{Z}/n\mathbb{Z}$ and $CH^2(Q) = (\mathbb{Z}/n\mathbb{Z})^{\oplus 2}$ (with $\mathbb{Z}/n\mathbb{Z}$ coefficients) it follows that $CH^2(A)$ is nonzero.

answered Sep 27 2011 at 9:16

ulrich
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	Thanks! Still, could you provide me with some references for this example? Also, are there any (natural) generalizations for it? From mine (motivic) point of view, this question should depend on the ground field; does it (for large $n$)? – Mikhail Bondarko Sep 28 2011 at 20:21
	Quadrics (over an algebraically closed field) have a cell decomposition which is what allows one to compute the Chow groups. There is a single cell in each dimension except for $d/2$ if the dimension $d$ is even, so this gives what I said. One can replace quadrics by Grassmannians to get similar examples. Of course, many more can be constructed using Damian Roessler's comment. – ulrich Sep 29 2011 at 6:13

An example of an affine variety with non-zero Chow groups

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