The Stacks project

Lemma 45.10.3. Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $X$ be a nonempty smooth projective scheme equidimensional of dimension $d$ over $k$. The diagram

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

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. Let $x$ be a closed point of $X$ whose residue field is separable over $k$. View $x$ as a scheme and denote $i : x \to X$ the inclusion morphism. To avoid confusion denote $\gamma ' : \mathop{\mathrm{CH}}\nolimits _0(x) \to H^0(x)$ the cycle class map for $x$. Then we have

\[ \int _ X \gamma ([x]) = \int _ X \gamma (i_*[x]) = \int _ X i_*\gamma '([x]) = \int _ x \gamma '([x]) = \deg (x \to \mathop{\mathrm{Spec}}(k)) \]

The second equality is axiom (C)(b) and the third equality is the definition of $i_*$ on cohomology. The final equality is Lemma 45.10.2. This proves the lemma because $\mathop{\mathrm{CH}}\nolimits _0(X)$ is generated by the classes of points $x$ as above by Lemma 45.8.1. $\square$

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