Skip to content
Permalink
Browse files

Adjust notation

  • Loading branch information...
aisejohan committed May 15, 2019
1 parent a369d19 commit c8db32322b255604bed0e1047c013c0590415984
Showing with 10 additions and 10 deletions.
  1. +10 −10 chow.tex
@@ -9023,15 +9023,15 @@ \section{Higher codimension gysin homomorphisms}
Section \ref{section-blowup-Z-first}. By Lemma \ref{lemma-gysin-at-infty}
we have a canonical bivariant class in
$$
c_1 \in A^0(W_\infty \to X)
C \in A^0(W_\infty \to X)
$$
Consider the open immersion $j : C_ZX \to W_\infty$ of
(\ref{item-cone-is-open}) and the closed immersion
$i : C_ZX \to N$ constructed above. By Lemma \ref{lemma-vectorbundle}
for every $\alpha \in A_k(X)$ there exists a unique
$\beta \in A_*(Z)$ such that
$$
i_*j^*(c_1 \cap \alpha) = p^*\beta
i_*j^*(C \cap \alpha) = p^*\beta
$$
If $\mathcal{N}$ has constant rank $r$, then $\beta \in A_{k - r}(Z)$.

@@ -9046,11 +9046,11 @@ \section{Higher codimension gysin homomorphisms}
\end{lemma}

\begin{proof}
Since both $i_* \circ j^* \circ c_1$ and $p^*$ are bivariant classes
Since both $i_* \circ j^* \circ C$ and $p^*$ are bivariant classes
(see Lemmas \ref{lemma-flat-pullback-bivariant} and
\ref{lemma-push-proper-bivariant}) we can use the equation
$$
i_* \circ j^* \circ c_1 = p^* \circ c(Z \to X, \mathcal{N})
i_* \circ j^* \circ C = p^* \circ c(Z \to X, \mathcal{N})
$$
(suitably interpreted) to define $c(Z \to X, \mathcal{N})$
as a bivariant class. This works because $p^*$ is always
@@ -9084,16 +9084,16 @@ \section{Higher codimension gysin homomorphisms}
W'_\infty \to W_\infty \times_X X',\quad
C_{Z'}X' \to C_ZX \times_Z Z'
$$
To get $c \cap \alpha'$ we use the class $c_1 \cap \alpha'$
To get $c \cap \alpha'$ we use the class $C \cap \alpha'$
defined using the morphism
$W \times_{\mathbf{P}^1_X} \mathbf{P}^1_{X'} \to \mathbf{P}^1_{X'}$
in Lemma \ref{lemma-gysin-at-infty}.
To get $c' \cap \alpha'$ on the other hand, we use the class
$c'_1 \cap \alpha'$ defined using the morphism $W' \to \mathbf{P}^1_{X'}$.
$C' \cap \alpha'$ defined using the morphism $W' \to \mathbf{P}^1_{X'}$.
By Lemma \ref{lemma-gysin-at-infty-independent} the pushforward of
$c'_1 \cap \alpha'$ by the closed immersion
$C' \cap \alpha'$ by the closed immersion
$W'_\infty \to (W \times_{\mathbf{P}^1_X} \mathbf{P}^1_{X'})_\infty$,
is equal to $c_1 \cap \alpha'$. Hence the same is true for the pullbacks
is equal to $C \cap \alpha'$. Hence the same is true for the pullbacks
to the opens
$$
C_{Z'}X' \subset W'_\infty,\quad
@@ -9153,11 +9153,11 @@ \section{Higher codimension gysin homomorphisms}
Let us trace through the steps in the definition of
$c(Z \to X, \mathcal{N}) \cap [X]_n$. Let $b : W \to \mathbf{P}^1_X$
be the blowing up of $\infty(Z)$. We first have to compute
$c_1 \cap [X]_n$ where $c_1 \in A^0(W_\infty \to X)$ is
$C \cap [X]_n$ where $C \in A^0(W_\infty \to X)$ is
the class of Lemma \ref{lemma-gysin-at-infty}.
To do this, note that $[W]_{n + 1}$
is a cycle on $W$ whose restriction to $\mathbf{A}^1_X$ is
equal to the flat pullback of $[X]_n$. Hence $c_1 \cap [X]_n$
equal to the flat pullback of $[X]_n$. Hence $C \cap [X]_n$
is equal to $i_\infty^*[W]_{n + 1}$. Since none of the components
of $W$ of $\delta$-dimension $n + 1$ is contained in $W_\infty$
we see that $i_\infty^*[W]_{n + 1} = [W_\infty \cap W]_n = [W_\infty]_n$.

0 comments on commit c8db323

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