Lemma 45.7.4. Let $H^*$ be a classical Weil cohomology theory (Definition 45.7.3). Let $X$ be a smooth projective variety of dimension $d$. The diagram

$\xymatrix{ \mathop{\mathrm{CH}}\nolimits ^ d(X) \ar[r]_-\gamma \ar@{=}[d] & H^{2d}(X) \ar[d]^{\int _ X} \\ \mathop{\mathrm{CH}}\nolimits _0(X) \ar[r]^\deg & F }$

commutes where $\deg : \mathop{\mathrm{CH}}\nolimits _0(X) \to \mathbf{Z}$ is the degree of zero cycles discussed in Chow Homology, Section 42.41.

Proof. The result holds for $\mathop{\mathrm{Spec}}(k)$ by axiom (C)(d). Let $x : \mathop{\mathrm{Spec}}(k) \to X$ be a closed point of $X$. Then we have $\gamma ([x]) = x_*\gamma ([\mathop{\mathrm{Spec}}(k)])$ in $H^{2d}(X)$ by axiom (C)(b). Hence $\int _ X \gamma ([x]) = 1$ by the definition of $x_*$. $\square$

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