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Improve statement + proof result of cb18842

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aisejohan committed May 20, 2019
1 parent cb18842 commit 96d32710998dfbfed757970cf81cb6109b1ea051
Showing with 27 additions and 32 deletions.
  1. +27 −32 chow.tex
@@ -9806,54 +9806,49 @@ \section{Higher codimension gysin homomorphisms}
\begin{lemma}
\label{lemma-gysin-commutes}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme
locally of finite type over $S$. Let $Z_i \subset X$ be closed subschemes
endowed with virtual normal sheaves $\mathcal{N}_i$. The bivariant
classes $c(Z_1 \to X, \mathcal{N}_1)$ and $c(Z_2 \to X, \mathcal{N}_2)$
commute in the sense of Remark \ref{remark-bivariant-commute}.
locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme
with virtual normal sheaf $\mathcal{N}$. Let $Y \to X$ be locally of
finite type and $c \in A^*(Y \to X)$. Then
$$
c \circ c(Z \to X, \mathcal{N}) =
c(Z \to X, \mathcal{N}) \circ c
$$
in $A^*(Z \times_X Y \to X)$.
\end{lemma}

\begin{proof}
Write $c_1 = c(Z_1 \to X, \mathcal{N}_1)$ and
$c_2 = c(Z_2 \to X, \mathcal{N}_2)$. To check
$res(c_1) \circ c_2 = res(c_2) \circ c_1$ in $A^*(Z_1 \cap Z_2 \to X)$
we use Lemma \ref{lemma-bivariant-zero}. Thus we may assume
$X$ is an integral scheme and we have to show
$c_1 \cap c_2 \cap [X] = c_2 \circ c_1 \cap [X]$ in $A^*(Z_1 \cap Z_2)$.
To check this we may use Lemma \ref{lemma-bivariant-zero}.
Thus we may assume $X$ is an integral scheme and we have to show
$c \cap c(Z \to X, \mathcal{N}) \cap [X] =
c(Z \to X, \mathcal{N}) \circ c \cap [X]$ in $A_*(Z \times_X Y)$.

\medskip\noindent
If $Z_1 = X$, then $c_1 = c_{top}(\mathcal{N}_1)$ by
If $Z = X$, then $c(Z \to X, \mathcal{N}) = c_{top}(\mathcal{N})$ by
Lemma \ref{lemma-gysin-fundamental} which commutes
with the bivariant class $c_2$, see Lemma \ref{lemma-cap-commutative-chern}.
The same argument works if $Z_2 = X$.
with the bivariant class $c$, see Lemma \ref{lemma-cap-commutative-chern}.

\medskip\noindent
Assume that both $Z_1$ and $Z_2$ are not equal to $X$.
By Lemma \ref{lemma-bivariant-zero} it even suffices to
prove the result after blowing up $X$ (in a nonzero ideal).
Let us blowup $X$ in the product of the ideal sheaves of
$Z_1$ and $Z_2$. This transforms both $Z_1$ and $Z_2$ into
effective Cartier divisors, see
Divisors, Lemmas
Assume that $Z$ is not equal to $X$. By Lemma \ref{lemma-bivariant-zero}
it even suffices to prove the result after blowing up $X$ (in a nonzero ideal).
Let us blowup $X$ in the ideal sheaf of $Z$. This reduces us to the case
where $Z$ is an effective Cartier divisor, see
Divisors, Lemma
\ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor},
\ref{divisors-lemma-blow-up-pullback-effective-Cartier}, and
\ref{divisors-lemma-blowing-up-two-ideals}.
Thus we may assume $Z_1$ and $Z_2$ are effective Cartier divisors.

\medskip\noindent
If $Z_1$ and $Z_2$ are effective Cartier divisors, then we have
If $Z$ is an effective Cartier divisor, then we have
$$
c(Z_i \to X, \mathcal{N}_i) =
c_{top}(\mathcal{E}_i) \circ i_i^*
c(Z \to X, \mathcal{N}) =
c_{top}(\mathcal{E}) \circ i^*
$$
where $i_i^*$ is the gysing homomorphism associated to $i_i : Z_i \to X$
and $\mathcal{E}_i$ is the dual of the kernel of
$\mathcal{N}_i^\vee \to \mathcal{C}_{Z_i/X}$, see
where $i^*$ is the gysing homomorphism associated to $i : Z \to X$
and $\mathcal{E}$ is the dual of the kernel of
$\mathcal{N}^\vee \to \mathcal{C}_{Z/X}$, see
Lemmas \ref{lemma-gysin-decompose} and \ref{lemma-gysin-agrees}.
Then we conclude because chern classes are in the center of the
bivariant ring (in the strong sense formulated in
Lemma \ref{lemma-cap-commutative-chern}) and
we have the commutation relation for gysin homomorphisms $i_i^*$ by
Remark \ref{example-gysin-commute}.
Lemma \ref{lemma-cap-commutative-chern}) and commute
with the gysin homomorphism $i^*$ by definition of bivariant classes.
\end{proof}


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