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{{mfd-mergeto|Chow group|Draft:Operational Chow ring|9 March 2018|Talk:Chow group}} {{mfd-mergeto|Chow group|Draft:Operational Chow ring|9 March 2018|Talk:Chow group}}


The basic question was whether there is a ]: The basic question was whether there is a ]:
:<math>A^*(X) \to \operatorname{H}^*(X, \mathbb{Z}).</math> :<math>A^*(X) \to \operatorname{H}^*(X, \mathbb{Z}).</math>
If ''X'' is smooth, such a map exists since <math>A^*(X)</math> is the usual ] of ''X''. {{harv|Totaro|2014}} has shown that rationally there is no such a map with good properties even if ''X'' is a ], roughly a variety admitting a cell decomposition. He also notes that Voevodsky's ] is "probably more useful " than the operational Chow ring for a singular scheme (§ 8 of loc. cit.) If ''X'' is smooth, such a map exists since <math>A^*(X)</math> is the usual ] of ''X''. {{harv|Totaro|2014}} has shown that rationally there is no such a map with good properties even if ''X'' is a ], roughly a variety admitting a cell decomposition. He also notes that Voevodsky’s ] is "probably more useful " than the operational Chow ring for a singular scheme (§ 8 of loc. cit.)




== References == == References ==
*W. Fulton, R. MacPherson, F. Sottile, and B. Sturmfels, 'Intersection theory on spherical varieties', J. Alg. Geom. 4 (1995), 181–193. *W. Fulton, R. MacPherson, F. Sottile, and B. Sturmfels, ‘Intersection theory on spherical varieties’, J. Alg. Geom. 4 (1995), 181–193.
*Totaro, *Totaro,


Revision as of 07:41, 9 September 2018

This page was nominated for deletion. The debate was closed on 9 March 2018 with a consensus to merge the content into the page Chow group. If you find that such action has not been taken promptly, please consider assisting in the merger instead of re-nominating the page for deletion. To discuss the merger, please use the destination page's talk page.

The basic question was whether there is a cycle map:

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

If X is smooth, such a map exists since A ∗ ( X ) {\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

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