The Stacks project

Proof. We will use without further mention that we've constructed our cycle class map $\gamma $ in Lemma 45.14.2 compatible with intersection products and pullbacks and that we've already shown axioms (A), (B), (C)(a), (C)(c), and (C)(d) of Section 45.9, see Lemma 45.14.5, Remark 45.14.6, and Lemmas 45.14.7 and 45.14.8. In particular, we may use (for example) Lemma 45.9.1 to see that pushforward on $H^*$ is compatible with composition and satisfies the projection formula.

Let $f : X \to Y$ be a morphism of nonempty equidimensional smooth projective schemes over $k$. We are trying to show $f_*\gamma (\alpha ) = \gamma (f_*\alpha )$ for any cycle class $\alpha $ on $X$. We can write $\alpha $ as a $\mathbf{Q}$-linear combination of products of Chern classes of locally free $\mathcal{O}_ X$-modules (Chow Homology, Lemma 42.58.1). Thus we may assume $\alpha $ is a product of Chern classes of finite locally free $\mathcal{O}_ X$-modules $\mathcal{E}_1, \ldots , \mathcal{E}_ r$. Pick $p : P \to X$ as in the splitting principle (Chow Homology, Lemma 42.43.1). By Chow Homology, Remark 42.43.2 we see that $p$ is a composition of projective space bundles and that $\alpha = p_*(\xi _1 \cap \ldots \cap \xi _ d \cap \cdot p^*\alpha )$ where $\xi _ i$ are first Chern classes of invertible modules. By Lemma 45.14.9 we know that $p_*$ commutes with cycle classes. Thus it suffices to prove the property for the composition $f \circ p$. Since $p^*\mathcal{E}_1, \ldots , p^*\mathcal{E}_ r$ have filtrations whose successive quotients are invertible modules, this reduces us to the case where $\alpha $ is of the form $\xi _1 \cap \ldots \cap \xi _ t \cap [X]$ for some first Chern classes $\xi _ i$ of invertible modules $\mathcal{L}_ i$.

Assume $\alpha = c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_ t) \cap [X]$ for some invertible modules $\mathcal{L}_ i$ on $X$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module. For $n \gg 0$ the invertible $\mathcal{O}_ X$-modules $\mathcal{L}^{\otimes n}$ and $\mathcal{L}_1 \otimes \mathcal{L}^{\otimes n}$ are both very ample on $X$ over $k$, see Morphisms, Lemma 29.39.8. Since $c_1(\mathcal{L}_1) = c_1(\mathcal{L}_1 \otimes \mathcal{L}^{\otimes n}) - c_1(\mathcal{L}^{\otimes n})$ this reduces us to the case where $\mathcal{L}_1$ is very ample. Repeating this with $\mathcal{L}_ i$ for $i = 2, \ldots , t$ we reduce to the case where $\mathcal{L}_ i$ is very ample on $X$ over $k$ for all $i = 1, \ldots , t$.

Assume $k$ is infinite and $\alpha = c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_ t) \cap [X]$ for some very ample invertible modules $\mathcal{L}_ i$ on $X$ over $k$. By Bertini in the form of Varieties, Lemma 33.47.3 we can successively find regular sections $s_ i$ of $\mathcal{L}_ i$ such that the schemes $Z(s_1) \cap \ldots \cap Z(s_ i)$ are smooth over $k$ and of codimension $i$ in $X$. By the construction of capping with the first class of an invertible module (going back to Chow Homology, Definition 42.24.1), this reduces us to the case where $\alpha = [Z]$ for some nonempty smooth closed subscheme $Z \subset X$ which is equidimensional.

Assume $\alpha = [Z]$ where $Z \subset X$ is a smooth closed subscheme. Choose a closed embedding $X \to \mathbf{P}^ n$. We can factor $f$ as

Since we know the result for the second morphism by Lemma 45.14.9 it suffices to prove the result when $\alpha = [Z]$ where $i : Z \to X$ is a closed immersion and $f$ is a closed immersion. Then $j = f \circ i$ is a closed embedding too. Using the hypothesis for $i$ and $j$ we win.

We still have to prove the lemma in case $k$ is finite. We urge the reader to skip the rest of the proof. Everything we said above continues to work, except that we do not know we can choose the sections $s_ i$ cutting out our $Z$ over $k$ as $k$ is finite. However, we do know that we can find $s_ i$ over the algebraic closure $\overline{k}$ of $k$ (by the same lemma). This means that we can find a finite extension $k'/k$ such that our sections $s_ i$ are defined over $k'$. Denote $\pi : X_{k'} \to X$ the projection. The arguments above shows that we get the desired conclusion (from the assumption in the lemma) for the cycle $\pi ^*\alpha $ and the morphism $f \circ \pi : X_{k'} \to Y$. We have $\pi _*\pi ^*\alpha = [k' : k] \alpha $, see Chow Homology, Lemma 42.15.2. On the other hand, we have

by the projection formula for our cohomology theory. Observe that $\pi $ is a projection (!) and hence we have $\pi _*(1) = \int _{\mathop{\mathrm{Spec}}(k')}(1) 1$ by Lemma 45.9.2. Thus to finish the proof in the finite field case, it suffices to prove that $\int _{\mathop{\mathrm{Spec}}(k')}(1) = [k' : k]$ which we do in Lemma 45.14.10. $\square$