stacks/stacks-project

Introducing motives

It turns out to focus the discussion of Weil cohomology theories
suitably. Fun!
 @@ -5917,6 +5917,39 @@ \section{Bivariant intersection theory} condition (3) of Definition \ref{definition-bivariant-class}. \end{example} \begin{remark} \label{remark-more-general-bivariant} There is a more general type of bivariant class that doesn't seem to be considered in the literature. Namely, suppose we are given a diagram $$X \longrightarrow Z \longleftarrow Y$$ of schemes locally of finite type over $(S, \delta)$ as in Situation \ref{situation-setup}. Let $p \in \mathbf{Z}$. Then we can consider a rule $c$ which assigns to every $Z' \to Z$ locally of finite type maps $$c \cap - : A_k(Y') \longrightarrow A_{k - p}(X')$$ for all $k \in \mathbf{Z}$ where $X' = Z' \times_Z X$ and $Y' = Z' \times_Z Y$ compatible with \begin{enumerate} \item proper pushforward if given $Z'' \to Z'$ proper, \item flat pullback if given $Z'' \to Z'$ flat of fixed relative dimension, and \item gysin maps if given $D' \subset Z'$ as in Definition \ref{definition-gysin-homomorphism}. \end{enumerate} We omit the detailed formulations. Suppose we denote the collection of all such operations $A^*(X \to Z \leftarrow Y)$. A simple example of the utility of this concept is when we have a proper morphism $f : X_2 \to X_1$. Then $f_*$ isn't a bivariant operation in the sense of Definition \ref{definition-bivariant-class} but it is in the above generalized sense, namely, $f_* \in A^0(X_1 \to X_1 \leftarrow X_2)$. \end{remark}