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I have seen two definitions of intersection product in Aglebraic Geometry so far: one using the Chow ring of a variety (as defined for example in 3264 and all that, by Eisenbud and Harris) and the other one using sheaf cohomology (as defined for example in Vakil's notes, def. 20.1.1., or in Hartshorne's Ample subvarieties of algebraic varieties).

How are these two definitions related?

If we take $D$ to be a nice effective divisor on our (smooth, projective...) variety $X$ and we take $Y$ to be an $s$-dimensional subvariety, do we have the following equality?

$$(D^s.Y)=\deg([D]^s [Y])$$

(The left hand side being the definition using sheaf cohomology and the right hand side the definition using algebraic cycles).

What if we take $s$ different divisors? Under what conditions do we have equality?

For example in the case of surfaces the two products should agree, right?

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In section 20.3 Ravi Vakil gives hints as how to relate those two approaches, namely he mentions a surjective map (which becomes an isomorphism after tensoring with $\mathbb{Q}$) $$A_d(X) \to A_{d}^{'}(X)$$where the left hand side group is the usual Chow group and on the right - the group adapted to his approach throughout chapter 20. He mentions Fulton's book on Intersection Theory for these facts, namely examples 15.1.5 and theorems 18.2 and 18.3.

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