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The basic question, still open as of, is whether there is a ]: The basic question, still open as of, is 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''.) If ''X'' is smooth, such a map exists since <math>A^*(X)</math> is the usual ] of ''X''. {{harv|Totaro|2014}} has shown that there might not be such a map even of ''X'' is a ] (a variety admitting a certain cell decomposition.)


== References == == References ==

Revision as of 04:53, 4 November 2015

The basic question, still open as of, is 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 there might not be such a map even of X is a linear variety (a variety admitting a certain cell decomposition.)

References

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