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| {{mfd-mergeto|Chow group|Draft:Operational Chow ring|9 March 2018|Talk:Chow group}} | |||
| #REDIRECT ] | |||
| The basic question was whether there is a ]: | |||
| {{r to section}} | |||
| :<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.) | |||
| == References == | |||
| *W. Fulton, R. MacPherson, F. Sottile, and B. Sturmfels, ‘Intersection theory on spherical varieties’, J. Alg. Geom. 4 (1995), 181–193. | |||
| *Totaro, | |||
| {{geometry-stub}} | |||
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The basic question was whether there is a cycle map:
If X is smooth, such a map exists since 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.)
is the usual
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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