Lemma 45.14.13. Assume given data (D0), (D1), and (D2') satisfying axioms (A1) – (A7). In order to show that $\gamma$ commutes with pushforward it suffices to show that $i_*(1) = \gamma ([Z])$ if $i : Z \to X$ is a closed immersion of nonempty smooth projective equidimensional schemes over $k$ such that the class of $\mathcal{C}_{Z/X}$ in $K_0(Z)$ is the pullback of a class in $K_0(X)$.

Proof. By Lemma 45.14.11 it suffices to show that $i_*(1) = \gamma ([Z])$ if $i : Z \to X$ is a closed immersion of nonempty smooth projective equidimensional schemes over $k$. Say $Z$ has codimension $r$ in $X$. Let $\mathcal{L}$ be a sufficiently ample invertible module on $X$. Choose $n > 0$ and a surjection

$\mathcal{O}_ Z^{\oplus n} \to \mathcal{C}_{Z/X} \otimes \mathcal{L}|_ Z$

This gives a morphism $g : Z \to \mathbf{G}(n - r, n)$ to the Grassmanian over $k$, see Constructions, Section 27.22. Consider the composition

$Z \to X \times \mathbf{G}(n - r, n) \to X$

Pushforward along the second morphism is compatible with classes of cycles by Lemma 45.14.12. The conormal sheaf $\mathcal{C}$ of the closed immersion $Z \to X \times \mathbf{G}(n - r, n)$ sits in a short exact sequence

$0 \to \mathcal{C}_{Z/X} \to \mathcal{C} \to g^*\Omega _{\mathbf{G}(n - r, n)} \to 0$

See More on Morphisms, Lemma 37.11.13. Since $\mathcal{C}_{Z/X} \otimes \mathcal{L}|_ Z$ is the pull back of a finite locally free sheaf on $\mathbf{G}(n - r, n)$ we conclude that the class of $\mathcal{C}$ in $K_0(Z)$ is the pullback of a class in $K_0(X \times \mathbf{G}(n - r, n))$. Hence we have the property for $Z \to X \times \mathbf{G}(n - r, n)$ and we conclude. $\square$

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