The Stacks project

Lemma 42.54.2. The construction above defines a bivariant class1

\[ c(Z \to X, \mathcal{N}) \in A^*(Z \to X)^\wedge \]

and moreover the construction is compatible with base change as in Lemma 42.54.1. If $\mathcal{N}$ has constant rank $r$, then $c(Z \to X, \mathcal{N}) \in A^ r(Z \to X)$.

Proof. Since both $i_* \circ j^* \circ C$ and $p^*$ are bivariant classes (see Lemmas 42.33.2 and 42.33.4) we can use the equation

\[ i_* \circ j^* \circ C = p^* \circ c(Z \to X, \mathcal{N}) \]

(suitably interpreted) to define $c(Z \to X, \mathcal{N})$ as a bivariant class. This works because $p^*$ is always bijective on chow groups by Lemma 42.36.3.

Let $X' \to X$, $Z' \to X'$, and $\mathcal{N}'$ be as in Lemma 42.54.1. Write $c = c(Z \to X, \mathcal{N})$ and $c' = c(Z' \to X', \mathcal{N}')$. The second statement of the lemma means that $c'$ is the restriction of $c$ as in Remark 42.33.5. Since we claim this is true for all $X'/X$ locally of finite type, a formal argument shows that it suffices to check that $c' \cap \alpha ' = c \cap \alpha '$ for $\alpha ' \in \mathop{\mathrm{CH}}\nolimits _ k(X')$. To see this, note that we have a commutative diagram

\[ \xymatrix{ C_{Z'}X' \ar[d] \ar[r] & W'_\infty \ar[d] \ar[r] & W' \ar[d] \ar[r] & \mathbf{P}^1_{X'} \ar[d] \\ C_ ZX \ar[r] & W_\infty \ar[r] & W \ar[r] & \mathbf{P}^1_ X } \]

which induces closed immersions:

\[ W' \to W \times _{\mathbf{P}^1_ X} \mathbf{P}^1_{X'},\quad W'_\infty \to W_\infty \times _ X X',\quad C_{Z'}X' \to C_ ZX \times _ Z Z' \]

To get $c \cap \alpha '$ we use the class $C \cap \alpha '$ defined using the morphism $W \times _{\mathbf{P}^1_ X} \mathbf{P}^1_{X'} \to \mathbf{P}^1_{X'}$ in Lemma 42.48.1. To get $c' \cap \alpha '$ on the other hand, we use the class $C' \cap \alpha '$ defined using the morphism $W' \to \mathbf{P}^1_{X'}$. By Lemma 42.48.3 the pushforward of $C' \cap \alpha '$ by the closed immersion $W'_\infty \to (W \times _{\mathbf{P}^1_ X} \mathbf{P}^1_{X'})_\infty $, is equal to $C \cap \alpha '$. Hence the same is true for the pullbacks to the opens

\[ C_{Z'}X' \subset W'_\infty ,\quad C_ ZX \times _ Z Z' \subset (W \times _{\mathbf{P}^1_ X} \mathbf{P}^1_{X'})_\infty \]

by Lemma 42.15.1. Since we have a commutative diagram

\[ \xymatrix{ C_{Z'} X' \ar[d] \ar[r] & N' \ar@{=}[d] \\ C_ ZX \times _ Z Z' \ar[r] & N \times _ Z Z' } \]

these classes pushforward to the same class on $N'$ which proves that we obtain the same element $c \cap \alpha ' = c' \cap \alpha '$ in $\mathop{\mathrm{CH}}\nolimits _*(Z')$. $\square$

[1] The notation $A^*(Z \to X)^\wedge $ is discussed in Remark 42.35.5. If $X$ is quasi-compact, then $A^*(Z \to X)^\wedge = A^*(Z \to X)$.

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