Draft:Operational Chow ring: Difference between revisions

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	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 ~~might~~ ~~not be~~ such a map 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 ==

Revision as of 21:41, 6 November 2015

The basic question was whether there is a cycle map:

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

If X is smooth, such a map exists since $A^{*}(X)$ 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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