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Introducing motives

It turns out to focus the discussion of Weil cohomology theories
suitably. Fun!
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aisejohan committed Jun 14, 2019
1 parent 1eb5372 commit c0f582b70e6b00cd2e49770ea32535a78d9cf2ef
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  1. +33 −0 chow.tex
  2. +2,599 −467 weil.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}






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