Exterior products in chow

chow.tex

@@ -9773,6 +9773,150 @@ \section{Higher codimension gysin homomorphisms}
 
 
 
+\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}.
+Consider a cartesian square
+$$
+\xymatrix{
+X \times_k Y \ar[r] \ar[d] & Y \ar[d] \\
+X \ar[r] & \Spec(k) = S
+}
+$$
+of schemes locally of finite type over $k$. Then there is a canonical map
+$$
+\times :
+A_n(X) \otimes_{\mathbf{Z}} A_m(Y)
+\longrightarrow
+A_{n + m}(X \times_k Y)
+$$
+which is uniquely determined by the following rule:
+given integral closed subschemes $X' \subset X$
+and $Y' \subset Y$ of dimensions $n$ and $m$ we have
+$$
+[X'] \times [Y'] = [X' \times_k Y']_{n + m}
+$$
+in $A_{n + m}(X \times_k Y)$.
+
+\begin{lemma}
+\label{lemma-exterior-product-well-defined}
+The map
+$\times : A_n(X) \otimes_{\mathbf{Z}} A_m(Y) \to A_{n + m}(X \times_k Y)$
+is well defined.
+\end{lemma}
+
+\begin{proof}
+A first remark is that if $\alpha = \sum n_i[X_i]$
+and $\beta = \sum m_j[Y_j]$ with $X_i \subset X$ and $Y_j \subset Y$
+locally finite families of integral closed subschemes of
+dimensions $n$ and $m$, then
+$X_i \times_k Y_j$ is a locally finite
+collection of closed subschemes of $X \times_k Y$ of
+dimensions $n + m$ and we can indeed consider
+$$
+\alpha \times \beta = \sum n_i m_j [X_i \times_k Y_j]_{n + m}
+$$
+as a $(n + m)$-cycle on $X \times_k Y$. In this way we obtain an
+additive map
+$\times : Z_n(X) \otimes_{\mathbf{Z}} Z_m(Y) \to Z_{n + m}(X \times_k Y)$.
+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 dimension $n$.
+Then flat pullback along the morphism $p' : X' \to \Spec(k)$ is an element
+$(p')^* \in A^{-n}(X' \to \Spec(k))$ by
+Lemma \ref{lemma-flat-pullback-bivariant}
+and hence $c' = i_* \circ (p')^* \in A^{-n}(X \to \Spec(k))$ by
+Lemma \ref{lemma-push-proper-bivariant}.
+This produces maps
+$$
+c' \cap - : A_m(Y) \longrightarrow A_{m + n}(X \times_k Y)
+$$
+which the reader easily sends $[Y']$ to $[X' \times_k Y']_{n + m}$
+for any integral closed subscheme $Y' \subset Y$ of dimension
+$m$. Hence the construction
+$([X'], [Y']) \mapsto [X' \times_k Y']_{n + m}$
+factors through rational equivalence in the second variable, i.e.,
+gives a well defined map
+$Z_n(X) \otimes_{\mathbf{Z}} A_m(Y) \to A_{n + m}(X \times_k Y)$.
+By symmetry the same is true for the other variable and we conclude.
+\end{proof}
+
+\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.
+Then we have a canonical identification
+$$
+A^p(X \to \Spec(k)) = A_{-p}(X)
+$$
+for all $p \in \mathbf{Z}$.
+\end{lemma}
+
+\begin{proof}
+Observe that $A_*(\Spec(k))$ is nonzero only in degree $d$
+with generator $[\Spec(k)]$. Hence we get a map
+$A^p(X \to \Spec(k)) \to A_{-p}(X)$ by sending $c$ to
+$c \cap [\Spec(k)]$.
+
+\medskip\noindent
+Conversely, suppose we have $\alpha \in A_{-p}(X)$.
+Then we can define $c_\alpha \in A^p(X \to \Spec(k))$ as
+follows: given $X' \to \Spec(k)$ and $\alpha' \in A_n(X')$
+we let
+$$
+c_\alpha \cap \alpha' = \alpha \times \alpha'
+$$
+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
+$$
+\coprod X_i \xrightarrow{g} X \to \Spec(k)
+$$
+and denote $f : \coprod X_i \to \Spec(k)$ the composition.
+Then $g$ is proper and $f$ is flat of relative dimension $-p$.
+Pullback along $f$ is a bivariant class
+$f^* \in A^p(\coprod X_i \to \Spec(k))$ by
+Lemma \ref{lemma-flat-pullback-bivariant}.
+Denote $c' \in A^0(\coprod X_i)$ the bivariant class
+which multiplies a cycle by $n_i$ on the $i$th component.
+Thus $c' \circ f^* \in A^p(\coprod X_i \to X)$.
+Finally, we have a bivariant class
+$$
+g_* \circ c' \circ f^*
+$$
+by Lemma \ref{lemma-push-proper-bivariant}. The reader easily
+verifies that $c_\alpha$ is equal to this class 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 $c_\alpha \cap [\Spec(k)] = \alpha$
+this is clear for one of the two directions. For the other, let
+$c \in A^p(X \to \Spec(k))$ and set $\alpha = c \cap [\Spec(k)]$.
+It suffices to prove that
+$$
+c \cap [X'] = c_\alpha \cap [X']
+$$
+when $X'$ is an integral scheme locally of finite type over $\Spec(k)$,
+see Lemma \ref{lemma-bivariant-zero}. However, then $p' : X' \to \Spec(k)$
+is flat of relative dimension $\dim(X')$ and hence
+$[X'] = (p')^*[\Spec(k)]$. Thus the fact that the bivariant classes
+$c$ and $c_\alpha$ agree on $[\Spec(k)]$ implies they
+agree when capped against $[X']$ and the proof is complete.
+\end{proof}
+
+
+
+
+
+
+