Introducing motives

It turns out to focus the discussion of Weil cohomology theories
suitably. Fun!

chow.tex

@@ -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}
+
+
+