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Intersection products for smooth / Dedekind

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aisejohan committed May 22, 2019
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  1. +193 −2 chow.tex
195 chow.tex
@@ -10149,7 +10149,7 @@ \section{Exterior product}



\section{Intersection products for smooth varieties}
\section{Intersection products}
\label{section-intersection-product}

\noindent
@@ -10291,7 +10291,7 @@ \section{Intersection products for smooth varieties}



\section{Exterior product over a $1$-dimensional base}
\section{Exterior product over Dedekind domains}
\label{section-exterior-product-dim-1}

\noindent
@@ -10500,6 +10500,197 @@ \section{Exterior product over a $1$-dimensional base}
Some details omitted.
\end{proof}

\begin{remark}
\label{remark-commuting-exterior-dim-1}
The upshot of Lemmas \ref{lemma-chow-cohomology-towards-base-dim-1}
and \ref{lemma-chow-cohomology-towards-base-dim-1-commutes} is the following.
Let $(S, \delta)$ be as above. Let $X$ be a scheme locally of finite type
over $S$.
Let $\alpha \in A_*(X)$. Let $Y \to Z$ be a morphism of schemes
locally of finite type over $S$. Let $c' \in A^q(Y \to Z)$. Then
$$
\alpha \times (c' \cap \beta) = c' \cap (\alpha \times \beta)
$$
in $A_*(X \times_S Y)$ for any $\beta \in A_*(Z)$. Namely, this
follows by taking $c = c_\alpha \in A^*(X \to S)$ the bivariant class
corresponding to $\alpha$, see proof of
Lemma \ref{lemma-chow-cohomology-towards-base-dim-1}.
\end{remark}

\begin{lemma}
\label{lemma-exterior-product-associative-dim-1}
Exterior product is associative. More precisely, let $(S, \delta)$ be
as above, let $X, Y, Z$ be schemes locally of finite type over $S$, let
$\alpha \in A_*(X)$, $\beta \in A_*(Y)$, $\gamma \in A_*(Z)$.
Then $(\alpha \times \beta) \times \gamma =
\alpha \times (\beta \times \gamma)$ in $A_*(X \times_S Y \times_S Z)$.
\end{lemma}

\begin{proof}
Omitted. Hint: associativity of fibre product of schemes.
\end{proof}















\section{Intersection products over Dedekind domains}
\label{section-intersection-product-dim-1}

\noindent
Let $S$ be a locally Noetherian scheme which has an open covering
by spectra of Dedekind domains. Set $\delta(s) = 0$ for $s \in S$ closed
and $\delta(s) = 1$ otherwise. Then $(S, \delta)$ is a special case of our
general Situation \ref{situation-setup}; see
Example \ref{example-domain-dimension-1} and discussion in
Section \ref{section-exterior-product-dim-1}.

\medskip\noindent
Let $X$ be a smooth scheme over $S$. The bivariant class $\Delta^!$
of Section \ref{section-gysin-for-diagonal} allows us to define a kind of
intersection product on chow groups of schemes locally of finite type over $X$.
Namely, suppose that $Y \to X$ and $Z \to X$ are morphisms of schemes
which are locally of finite type. Then observe that
$$
Y \times_X Z = (Y \times_S Z) \times_{X \times_S X, \Delta} X
$$
Hence we can consider the following sequence of maps
$$
A_n(Y) \otimes_\mathbf{Z} A_m(Y)
\xrightarrow{\times}
A_{n + m - 1}(Y \times_S Z)
\xrightarrow{\Delta^!}
A_{n + m - *}(Y \times_X Z)
$$
Here the first arrow is the exterior product constructed in
Section \ref{section-exterior-product-dim-1}. If $X$ is equidimensional
of dimension $d$, then $X \to S$ is smooth of relative dimension $d - 1$
and hence we end up in $A_{n + m - d}(Y \times_X Z)$.
In general we can decompose into the parts lying over the open
and closed subschemes of $X$ where $X$ has a given dimension.
Given $\alpha \in A_*(Y)$ and $\beta \in A_*(Z)$ we will denote
$$
\alpha \cdot \beta = \Delta^!(\alpha \times \beta)
\in A_*(Y \times_X Z)
$$
In the special case where $X = Y = Z$ we obtain a multiplication
$$
A_*(X) \times A_*(X) \to A_*(X),\quad
(\alpha, \beta) \mapsto \alpha \cdot \beta
$$
which is called the {\it intersection product}. We observe that
this product is clearly symmetric. Associativity follows from
the next lemma (as well as the one following).

\begin{lemma}
\label{lemma-associative-dim-1}
The product defined above is associative. More precisely, with
$(S, \delta)$ as above, let $X$ be smooth over $S$,
let $Y, Z, W$ be schemes locally of finite type over $X$, let
$\alpha \in A_*(Y)$, $\beta \in A_*(Z)$, $\gamma \in A_*(W)$.
Then $(\alpha \cdot \beta) \cdot \gamma =
\alpha \cdot (\beta \cdot \gamma)$ in $A_*(Y \times_X Z \times_X W)$.
\end{lemma}

\begin{proof}
By Lemma \ref{lemma-exterior-product-associative-dim-1} we have
$(\alpha \times \beta) \times \gamma =
\alpha \times (\beta \times \gamma)$ in $A_*(Y \times_S Z \times_S W)$.
Consider the closed immersions
$$
\Delta_{12} : X \times_S X \longrightarrow X \times_S X \times_S X,
\quad (x, x') \mapsto (x, x, x')
$$
and
$$
\Delta_{23} : X \times_S X \longrightarrow X \times_S X \times_S X,
\quad (x, x') \mapsto (x, x', x')
$$
Denote $\Delta_{12}^!$ and $\Delta_{23}^!$ the corresponding bivariant
classes; observe that $\Delta_{12}^!$ is the restriction
(Remark \ref{remark-restriction-bivariant}) of $\Delta^!$
to $X \times_S X \times_S X$ by the map $\text{pr}_{12}$ and that
$\Delta_{23}^!$ is the restriction of $\Delta^!$
to $X \times_S X \times_S X$ by the map $\text{pr}_{23}$.
Thus clearly the restriction of $\Delta_{12}^!$ by $\Delta_{23}$
is $\Delta^!$ and the restriction of $\Delta_{23}^!$ by $\Delta_{12}$ is
$\Delta^!$ too. Thus by Lemma \ref{lemma-gysin-commutes} we have
$$
\Delta^! \circ \Delta_{12}^! =
\Delta^! \circ \Delta_{23}^!
$$
Now we can prove the lemma by the following sequence of equalities:
\begin{align*}
(\alpha \cdot \beta) \cdot \gamma
& =
\Delta^!(\Delta^!(\alpha \times \beta) \times \gamma) \\
& =
\Delta^!(\Delta_{12}^!((\alpha \times \beta) \times \gamma)) \\
& =
\Delta^!(\Delta_{23}^!((\alpha \times \beta) \times \gamma)) \\
& =
\Delta^!(\Delta_{23}^!(\alpha \times (\beta \times \gamma)) \\
& =
\Delta^!(\alpha \times \Delta^!(\beta \times \gamma)) \\
& =
\alpha \cdot (\beta \cdot \gamma)
\end{align*}
All equalities are clear from the above except perhaps
for the second and penultimate one. The equation
$\Delta_{23}^!(\alpha \times (\beta \times \gamma)) =
\alpha \times \Delta^!(\beta \times \gamma)$ holds by
Remark \ref{remark-commuting-exterior}. Similarly for the second
equation.
\end{proof}

\begin{lemma}
\label{lemma-identify-chow-for-smooth}
Let $(S, \delta)$ be as above. Let $X$ be a smooth scheme over $S$,
equidimensional of dimension $d$. The map
$$
A^p(X) \longrightarrow A_{d - p}(X),\quad
c \longmapsto c \cap [X]_d
$$
is an isomorphism. Via this isomorphism composition of bivariant
classes turns into the intersection product defined above.
\end{lemma}

\begin{proof}
Denote $g : X \to S$ the structure morphism.
The map is the composition of the isomorphisms
$$
A^p(X) \to A^{p - d + 1}(X \to S) \to A_{d - p}(X)
$$
The first is the isomorphism $c \mapsto c \circ g^*$ of
Proposition \ref{proposition-compute-bivariant}
and the second is the isomorphism $c \mapsto c \cap [S]_1$ of
Lemma \ref{lemma-chow-cohomology-towards-base-dim-1}.
From the proof of Lemma \ref{lemma-chow-cohomology-towards-base-dim-1}
we see that the inverse to the second arrow sends $\alpha \in A_{d - p}(X)$
to the bivariant class $c_\alpha$ which sends $\beta \in A_*(Y)$
for $Y$ locally of finite type over $k$
to $\alpha \times \beta$ in $A_*(X \times_k Y)$. From the proof of
Proposition \ref{proposition-compute-bivariant} we see the inverse
to the first arrow in turn sends $c_\alpha$ to the bivariant class
which sends $\beta \in A_*(Y)$ for $Y \to X$ locally of finite type
to $\Delta^!(\alpha \times \beta) = \alpha \cdot \beta$.
From this the final result of the lemma follows.
\end{proof}








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