Skip to content
Permalink
Browse files

Another commutation relation

  • Loading branch information...
aisejohan committed May 21, 2019
1 parent 82047c0 commit b5c02b3f54dafcde7bf693c55d04f63899bb9885
Showing with 41 additions and 7 deletions.
  1. +41 −7 chow.tex
@@ -7193,7 +7193,7 @@ \section{Chern classes and sections}
Remark \ref{remark-top-chern-class}. By its very definition, in
order to verify the formula, we may assume that $\mathcal{E}$
has constant rank. We may similarly assume $\mathcal{N}'$ and
$\mathcal{N}$ have constant ranks, say $r$ and $r'$, so
$\mathcal{N}$ have constant ranks, say $r'$ and $r$, so
$\mathcal{E}$ has rank $r - r'$ and
$c_{top}(\mathcal{E}) = c_{r - r'}(\mathcal{E})$.
Observe that $p^*\mathcal{E}$ has a canonical section
@@ -7208,9 +7208,9 @@ \section{Chern classes and sections}
subscheme of $\delta$-dimension $n$. Then we have
\begin{enumerate}
\item $p^*[Y] = [p^{-1}(Y)]$ since $p^{-1}(Y)$ is integral of
dimension $n + r$,
$\delta$-dimension $n + r$,
\item $(p')^*[Y] = [(p')^{-1}(Y)]$ since $(p')^{-1}(Y)$ is integral of
dimension $n + r'$,
$\delta$-dimension $n + r'$,
\item the restriction of $s$ to $p^{-1}Y$ has vanishing scheme
$(p')^{-1}Y$ and the closed immersion $(p')^{-1}Y \to p^{-1}Y$
is a regular immersion (locally cut out by a regular sequence).
@@ -9598,7 +9598,7 @@ \section{Higher codimension gysin homomorphisms}
exact sequence $0 \to \mathcal{N}' \to \mathcal{N} \to \mathcal{E} \to 0$
of finite locally free $\mathcal{O}_Z$-modules such that the given surjection
$\sigma : \mathcal{N}^\vee \to \mathcal{C}_{Z/X}$ factors through a map
$\sigma' : (\mathcal{N}')^\vee \to \mathcal{C}_{Z/x}$.
$\sigma' : (\mathcal{N}')^\vee \to \mathcal{C}_{Z/X}$.
Then
$$
c(Z \to X, \mathcal{N}) = c_{top}(\mathcal{E}) \circ c(Z \to X, \mathcal{N}')
@@ -9882,12 +9882,12 @@ \section{Exterior product}
Pullback along $f$ is a bivariant class
$f^* \in A^p(\coprod X_i \to \Spec(k))$ by
Lemma \ref{lemma-flat-pullback-bivariant}.
Denote $c' \in A^0(\coprod X_i)$ the bivariant class
Denote $\nu \in A^0(\coprod X_i)$ the bivariant class
which multiplies a cycle by $n_i$ on the $i$th component.
Thus $c' \circ f^* \in A^p(\coprod X_i \to X)$.
Thus $\nu \circ f^* \in A^p(\coprod X_i \to X)$.
Finally, we have a bivariant class
$$
g_* \circ c' \circ f^*
g_* \circ \nu \circ f^*
$$
by Lemma \ref{lemma-push-proper-bivariant}. The reader easily
verifies that $c_\alpha$ is equal to this class and hence
@@ -9910,6 +9910,23 @@ \section{Exterior product}
agree when capped against $[X']$ and the proof is complete.
\end{proof}

\begin{lemma}
\label{lemma-chow-cohomology-towards-point-commutes}
Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$.
Let $c \in A^p(X \to \Spec(k))$. Let $Y \to Z$ be a morphism of schemes
locally of finite type over $k$. Let $c' \in A^q(Y \to Z)$. Then
$c \circ c' = c' \circ c$ in $A^{p + q}(X \times_k Y \to X \times_k Z)$.
\end{lemma}

\begin{proof}
In the proof of Lemma \ref{lemma-chow-cohomology-towards-point}
we have seen that $c$ is given by a combination of
proper pushforward, multiplying by integers over connected
components, and flat pullback. Since $c'$ commutes with each of
these operations by definition of bivariant classe, we conclude.
Some details omitted.
\end{proof}




@@ -10106,6 +10123,23 @@ \section{Exterior product over a $1$-dimensional base}
commute with flat pullback and gysin maps) and the proof is complete.
\end{proof}

\begin{lemma}
\label{lemma-chow-cohomology-towards-base-dim-1-commutes}
Let $(S, \delta)$ be as above. Let $X$ be a scheme locally of finite type
over $S$. Let $c \in A^p(X \to S)$. Let $Y \to Z$ be a morphism of schemes
locally of finite type over $S$. Let $c' \in A^q(Y \to Z)$. Then
$c \circ c' = c' \circ c$ in $A^{p + q}(X \times_S Y \to X \times_S Z)$.
\end{lemma}

\begin{proof}
In the proof of Lemma \ref{lemma-chow-cohomology-towards-base-dim-1}
we have seen that $c$ is given by a combination of
proper pushforward, multiplying by integers over connected
components, flat pullback, and gysin maps. Since $c'$ commutes with each of
these operations by definition of bivariant classes, we conclude.
Some details omitted.
\end{proof}




0 comments on commit b5c02b3

Please sign in to comment.
You can’t perform that action at this time.