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The basic question was whether there is a cycle map:

${\displaystyle A^{*}(X)\to \operatorname {H} ^{*}(X,\mathbb {Z} ).}$

If X is smooth, such a map exists since ${\displaystyle A^{*}(X)}$ is the usual Chow ring of X. (Totaro 2014) harv error: no target: CITEREFTotaro2014 (help) has shown that rationally there is no such a map with good properties even if X is a linear variety, roughly a variety admitting a cell decomposition. He also notes that Voevodsky’s motivic cohomology ring is "probably more useful " than the operational Chow ring for a singular scheme (§ 8 of loc. cit.)

References

• W. Fulton, R. MacPherson, F. Sottile, and B. Sturmfels, ‘Intersection theory on spherical varieties’, J. Alg. Geom. 4 (1995), 181–193.
• Totaro, Chow groups, Chow cohomology and linear varieties

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