The Stacks project

Remark 42.34.7. Let $(S, \delta )$ be as in Situation 42.7.1. Let $Z \to X$ be a closed immersion of schemes locally of finite type over $S$ and let $p \geq 0$. In this setting we define

\[ A^{(p)}(Z \to X) = \prod \nolimits _{i \leq p - 1} A^ i(X) \times \prod \nolimits _{i \geq p} A^ i(Z \to X). \]

Then $A^{(p)}(Z \to X)$ canonically comes equipped with the structure of a graded algebra. In fact, more generally there is a multiplication

\[ A^{(p)}(Z \to X) \times A^{(q)}(Z \to X) \longrightarrow A^{(\max (p, q))}(Z \to X) \]

In order to define these we define maps

\begin{align*} A^ i(Z \to X) \times A^ j(X) & \to A^{i + j}(Z \to X) \\ A^ i(X) \times A^ j(Z \to X) & \to A^{i + j}(Z \to X) \\ A^ i(Z \to X) \times A^ j(Z \to X) & \to A^{i + j}(Z \to X) \end{align*}

For the first we use composition of bivariant classes. For the second we use restriction $A^ i(X) \to A^ i(Z)$ (Remark 42.33.5) and composition $A^ i(Z) \times A^ j(Z \to X) \to A^{i + j}(Z \to X)$. For the third, we send $(c, c')$ to $res(c) \circ c'$ where $res : A^ i(Z \to X) \to A^ i(Z)$ is the restriction map (see Remark 42.33.5). We omit the verification that these multiplications are associative in a suitable sense.

Comments (0)

There are also:

  • 2 comment(s) on Section 42.34: Chow cohomology and the first Chern class

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Your email address will not be published.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 0FA0. Beware of the difference between the letter 'O' and the digit '0'.