Lemma 45.7.4 (0FGW)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 45: Weil Cohomology Theories
Section 45.7: Classical Weil cohomology theories
Lemma 45.7.4 (cite)

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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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