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 $A^*(Z \to X)^\wedge = A^*(Z \to X)$.

Comments (0)

There are also:

  • 1 comment(s) on Section 42.54: Higher codimension gysin homomorphisms

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FBK. Beware of the difference between the letter 'O' and the digit '0'.