Let $X$ be a smooth quasi-projective algebraic variety over $\mathbb{C}$ and $A^k(X)$ be the Chow group of codimension-$k$ algebraic cycles on $X$. let $\mathrm{cl}$ be the cycle map from $A^k(X)$ into the De Rham cohomology group $H^{2k}(X,\mathbb{C})$, given by the Poincare duality. Then, does the intersection product/pairing between the Chow groups correspond to the cup product (wedge product) between De Rham cohomology groups?

The answer is yes when $X$ is projective. It perhaps is also true when $X$ is quasi-projective. Does anyone know a reference or an argument that deals with the quasi-projective case?

(BCnrd briefly mentioned in another thread that for the analogue in etale cohomology, the conditions of properness and smoothness can both be removed, but the technique for proving the statement is different. Does anyone know a reference for this or which technique the comment refers to? )


It's certainly true. See corollary 19.2 in Fulton's Intersection Theory. In general, that's a good reference for a lot of these sorts of questions.

  • $\begingroup$ Thanks a lot. (For problems related to nonproper varieties and noncompact manifolds, I often wonder whether the theorems and constructions for compact varieties/manifolds would still work.) $\endgroup$ – John Apr 27 '14 at 0:37

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