In the book 3264 and All That by Eisenbud & Harris, the authors claim that for smooth projective varieties admitting an affine stratification, the algebraic equivalence relation and the rational equivalence relation define the same intersection theory (p. 553). Anyway, they do not give an explicit reference where one can find the proof of this claim. I am looking for such a reference.
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9$\begingroup$ In fact more is true: the cycle class map $CH^{*}(X)\rightarrow H^{*}(X)$ is an isomorphism. See e.g. Fulton's Intersection theory, Examples 1.9.1 and 19.1.11 (b). $\endgroup$– abxCommented Jan 22, 2019 at 19:32
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$\begingroup$ Many thanks for the answer! $\endgroup$– Vincenzo ZaccaroCommented Jan 22, 2019 at 21:18
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$\begingroup$ @abx: You might consider posting your above comment as an answer so that the question counts as being answered in the system. $\endgroup$– J WCommented Jan 11, 2021 at 14:42
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$\begingroup$ @J W: OK, just done. $\endgroup$– abxCommented Jan 12, 2021 at 15:23
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In fact more is true: the cycle class map $CH^*(X)\rightarrow H^*(X)$ is an isomorphism. See e.g. Fulton's Intersection theory, Examples 1.9.1 and 19.1.11 (b).