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Exterior product over dim 1 base

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aisejohan committed May 21, 2019
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  1. +199 −9 chow.tex
208 chow.tex
@@ -9777,8 +9777,9 @@ \section{Exterior product}
\label{section-exterior-product}

\noindent
Let $k$ be a field and let $(S, \delta)$ be as in Example \ref{example-field}
which is a special case of our general Situation \ref{situation-setup}.
Let $k$ be a field. Set $S = \Spec(k)$ and define $\delta : S \to \mathbf{Z}$
by sending the unique point to $0$. Then $(S, \delta)$ is a special case of
our general Situation \ref{situation-setup}, see Example \ref{example-field}.
Consider a cartesian square
$$
\xymatrix{
@@ -9849,8 +9850,7 @@ \section{Exterior product}

\begin{lemma}
\label{lemma-chow-cohomology-towards-point}
Let $k$ be a field and let $(S, \delta)$ be as in Example \ref{example-field}.
Let $X \to \Spec(k) \to S$ be morphisms locally of finite type.
Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$.
Then we have a canonical identification
$$
A^p(X \to \Spec(k)) = A_{-p}(X)
@@ -9859,10 +9859,9 @@ \section{Exterior product}
\end{lemma}

\begin{proof}
Observe that $A_*(\Spec(k))$ is nonzero only in degree $d$
Observe that $A_*(\Spec(k))$ is nonzero only in degree $0$
with generator $[\Spec(k)]$. Hence we get a map
$A^p(X \to \Spec(k)) \to A_{-p}(X)$ by sending $c$ to
$c \cap [\Spec(k)]$.
$A^p(X \to \Spec(k)) \to A_{-p}(X)$ by sending $c$ to $c \cap [\Spec(k)]$.

\medskip\noindent
Conversely, suppose we have $\alpha \in A_{-p}(X)$.
@@ -9873,8 +9872,8 @@ \section{Exterior product}
c_\alpha \cap \alpha' = \alpha \times \alpha'
$$
in $A_{n - p}(X \times_k X')$. To show that this is a bivariant
class we write $\alpha = \sum n_i[X_i]$ as a locally finite sum
of cycles on $X$. Consider the composition
class we write $\alpha = \sum n_i[X_i]$ as in
Definition \ref{definition-cycles}. Consider the composition
$$
\coprod X_i \xrightarrow{g} X \to \Spec(k)
$$
@@ -9915,6 +9914,197 @@ \section{Exterior product}



\section{Exterior product over a $1$-dimensional base}
\label{section-exterior-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}.
Observe that $S$ is normal
(Algebra, Lemma \ref{algebra-lemma-characterize-Dedekind})
and hence a disjoint union of normal integral schemes
(Properties, Lemma \ref{properties-lemma-normal-locally-Noetherian}).
Thus all of the arguments below reduce to the case where $S$ is
irreducible. On the other hand, we allow $S$ to be nonseparated (so $S$
could be the affine line with $0$ doubled for example).

\medskip\noindent
Consider a cartesian square
$$
\xymatrix{
X \times_S Y \ar[r] \ar[d] & Y \ar[d] \\
X \ar[r] & S
}
$$
of schemes locally of finite type over $S$. We claim there is a canonical map
$$
\times :
A_n(X) \otimes_{\mathbf{Z}} A_m(Y)
\longrightarrow
A_{n + m - 1}(X \times_S Y)
$$
which is uniquely determined by the following rule:
given integral closed subschemes $X' \subset X$
and $Y' \subset Y$ of $\delta$-dimensions $n$ and $m$ we set
\begin{enumerate}
\item $[X'] \times [Y'] = [X' \times_S Y']_{n + m - 1}$ if
$X'$ or $Y'$ dominates an irreducible component of $S$,
\item $[X'] \times [Y'] = 0$ if neither $X'$ nor $Y'$ dominates an
irreducible component of $S$.
\end{enumerate}

\begin{lemma}
\label{lemma-exterior-product-well-defined-dim-1}
The map
$\times : A_n(X) \otimes_{\mathbf{Z}} A_m(Y) \to A_{n + m - 1}(X \times_S Y)$
is well defined.
\end{lemma}

\begin{proof}
Consider $n$ and $m$ cycles $\alpha = \sum_{i \in I} n_i[X_i]$
and $\beta = \sum_{j \in J} m_j[Y_j]$ with $X_i \subset X$ and $Y_j \subset Y$
locally finite families of integral closed subschemes of
$\delta$-dimensions $n$ and $m$. Let $K \subset I \times J$ be the set
of pairs $(i, j) \in I \times J$ such that $X_i$ or $Y_j$ dominates
an irreducible component of $S$.
Then $\{X_i \times_S Y_j\}_{(i, j) \in K}$ is a locally finite
collection of closed subschemes of $X \times_S Y$ of
$\delta$-dimension $n + m - 1$. This means we can indeed consider
$$
\alpha \times \beta =
\sum\nolimits_{(i, j) \in K} n_i m_j [X_i \times_S Y_j]_{n + m - 1}
$$
as a $(n + m - 1)$-cycle on $X \times_S Y$. In this way we obtain an
additive map
$\times : Z_n(X) \otimes_{\mathbf{Z}} Z_m(Y) \to Z_{n + m}(X \times_S Y)$.
The problem is to show that
this procedure is compatible with rational equivalence.

\medskip\noindent
Let $i : X' \to X$ be the inclusion morphism of an integral closed subscheme
of $\delta$-dimension $n$ which dominates an irreducible component
of $S$. Then $p' : X' \to S$ is flat of relative dimension $n - 1$, see
More on Algebra, Lemma \ref{more-algebra-lemma-dedekind-torsion-free-flat}.
Hence flat pullback along $p'$ is an element
$(p')^* \in A^{-n + 1}(X' \to S)$ by
Lemma \ref{lemma-flat-pullback-bivariant}
and hence $c' = i_* \circ (p')^* \in A^{-n + 1}(X \to S)$ by
Lemma \ref{lemma-push-proper-bivariant}.
This produces maps
$$
c' \cap - : A_m(Y) \longrightarrow A_{m + n - 1}(X \times_S Y)
$$
which sends $[Y']$ to $[X' \times_S Y']_{n + m - 1}$ for any
integral closed subscheme $Y' \subset Y$ of $\delta$-dimension $m$.

\medskip\noindent
Let $i : X' \to X$ be the inclusion morphism of an integral closed subscheme
of $\delta$-dimension $n$ such that the composition $X' \to X \to S$
factors through a closed point $s \in S$. Since $s$ is a closed point
of the spectrum of a Dedekind domain, we see that $s$ is an effective
Cartier divisor on $S$ whose normal bundle is trivial. Denote
$c \in A^1(s \to S)$ the gysin homomorphism, see
Lemma \ref{lemma-gysin-bivariant}. The morphism $p' : X' \to s$
is flat of relative dimension $n$. Hence flat pullback along $p'$
is an element $(p')^* \in A^{-n}(X' \to S)$ by
Lemma \ref{lemma-flat-pullback-bivariant}.
Thus
$$
c' = i_* \circ (p')^* \circ c \in A^{-n}(X \to S)
$$
by Lemma \ref{lemma-push-proper-bivariant}. This produces maps
$$
c' \cap - : A_m(Y) \longrightarrow A_{m + n - 1}(X \times_S Y)
$$
which for any integral closed subscheme $Y' \subset Y$
of $\delta$-dimension $m$
sends $[Y']$ to either $[X' \times_S Y']_{n + m - 1}$ if $Y'$ dominates
an irreducible component of $S$ or to $0$ if not.

\medskip\noindent
From the previous two paragraphs we conclude
the construction $([X'], [Y']) \mapsto [X' \times_S Y']_{n + m - 1}$
factors through rational equivalence in the second variable, i.e.,
gives a well defined map
$Z_n(X) \otimes_{\mathbf{Z}} A_m(Y) \to A_{n + m - 1}(X \times_S Y)$.
By symmetry the same is true for the other variable and we conclude.
\end{proof}

\begin{lemma}
\label{lemma-chow-cohomology-towards-base-dim-1}
Let $(S, \delta)$ be as above. Let $X$ be a scheme locally of finite type
over $S$. Then we have a canonical identification
$$
A^p(X \to S) = A_{1 - p}(X)
$$
for all $p \in \mathbf{Z}$.
\end{lemma}

\begin{proof}
Consider the element $[S]_1 \in A_1(S)$. We get a map
$A^p(X \to S) \to A_{1 - p}(X)$ by sending $c$ to $c \cap [S]_1$.

\medskip\noindent
Conversely, suppose we have $\alpha \in A_{1 - p}(X)$.
Then we can define $c_\alpha \in A^p(X \to S)$ as
follows: given $X' \to S$ and $\alpha' \in A_n(X')$
we let
$$
c_\alpha \cap \alpha' = \alpha \times \alpha'
$$
in $A_{n - p}(X \times_S X')$. To show that this is a bivariant
class we write $\alpha = \sum_{i \in I} n_i[X_i]$ as in
Definition \ref{definition-cycles}. In particular the morphism
$$
g : \coprod\nolimits_{i \in I} X_i \longrightarrow X
$$
is proper. Pick $i \in I$. If $X_i$ dominates an irreducible component
of $S$, then the structure morphism $p_i : X_i \to S$ is flat and we have
$\xi_i = p_i^* \in A^p(X_i \to S)$. On the other hand, if $p_i$ factors
as $p'_i : X_i \to s_i$ followed by the inclusion $s_i \to S$
of a closed point, then we have
$\xi_i = (p'_i)^* \circ c_i \in A^p(X_i \to S)$
where $c_i \in A^1(s_i \to S)$ is the gysin homomorphism and
$(p'_i)^*$ is flat pullback. Observe that
$$
A^p(\coprod\nolimits_{i \in I} X_i \to S) =
\prod\nolimits_{i \in I} A^p(X_i \to S)
$$
Thus we have
$$
\xi = \sum n_i \xi_i \in A^p(\coprod\nolimits_{i \in I} X_i \to S)
$$
Finally, since $g$ is proper we have a bivariant class
$$
g_* \circ \xi \in A^p(X \to S)
$$
by Lemma \ref{lemma-push-proper-bivariant}. The reader easily
verifies that $c_\alpha$ is equal to this class
(please compare with the proof of
Lemma \ref{lemma-exterior-product-well-defined-dim-1})
and hence is itself a bivariant class.

\medskip\noindent
To finish the proof we have to show that the two constructions
are mutually inverse. Since $c_\alpha \cap [S]_1 = \alpha$
this is clear for one of the two directions. For the other, let
$c \in A^p(X \to S)$ and set $\alpha = c \cap [S]_1$.
It suffices to prove that
$$
c \cap [X'] = c_\alpha \cap [X']
$$
when $X'$ is an integral scheme locally of finite type over $S$,
see Lemma \ref{lemma-bivariant-zero}. However, either $p' : X' \to S$
is flat of relative dimension $\dim_\delta(X') - 1$ and hence
$[X'] = (p')^*[S]_1$ or $X' \to S$ factors as $X' \to s \to S$
and hence $[X'] = (p')^*(s \to S)^*[S]_1$. Thus the fact that the
bivariant classes $c$ and $c_\alpha$ agree on $[S]_1$
implies they agree when capped against $[X']$ (since bivariant classes
commute with flat pullback and gysin maps) and the proof is complete.
\end{proof}



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