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.

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