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I would like to know why for a smooth projective variety $X$ over an algebraically closed field $k$, numerical and homological equivalence coincide for divisors. Here by homological equivalence I mean that we have chosen a Weil cohomology theory with coefficients in a field $L$, in particular, there is no torsion.

What is a good reference for this.


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up vote 3 down vote accepted

There's a quick proof in Yves André's book "Une introduction aux motifs" (proposition

Note that a stronger result is true : actually, algebraic equivalence coincides with numerical equivalence for divisors on $X$ (*)("Matsusaka's theorem"). The only reference I know for this result is Matsusaka's original article ("The criteria for algebraic equivalence and the torsion group", Amer. J. Math. 79 (1957), 53–66), and I don't know if you would consider it a good reference.

(*) If your coefficients contain $\mathbb{Q}$, that is. If you take coefficients in $\mathbb{Z}$, then the result is that the group of divisors algebraically equivalent to zero is of finite index in the group of divisors numerically equivalent to zero.

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