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Gysin maps and excess intersection

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aisejohan committed Nov 27, 2019
1 parent 9155485 commit 8b678e5eaaee1501142dafa2d6663ec185d92e51
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  1. +109 −1 chow.tex
110 chow.tex
@@ -10919,6 +10919,44 @@ \section{Higher codimension gysin homomorphisms}
immediately from Lemma \ref{lemma-easy-virtual-class}.
\end{proof}
\begin{lemma}
\label{lemma-gysin-excess}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Consider
a cartesian diagram
$$
\xymatrix{
Z' \ar[r] \ar[d]_g & X' \ar[d]^f \\
Z \ar[r] & X
}
$$
of schemes locally of finite type over $S$ whose horizontal arrows
are closed immersions. Let $\mathcal{N}$, resp.\ $\mathcal{N}'$
be a virtual normal sheaf for $Z \subset X$, resp.\ $Z' \to X'$.
Assume given a short exact sequence
$0 \to \mathcal{N}' \to g^*\mathcal{N} \to \mathcal{E} \to 0$
of finite locally free modules on $Z'$ such that the diagram
$$
\xymatrix{
g^*\mathcal{N}^\vee \ar[r] \ar[d] &
(\mathcal{N}')^\vee \ar[d] \\
g^*\mathcal{C}_{Z/X} \ar[r] &
\mathcal{C}_{Z'/X'}
}
$$
commutes. Then we have
$$
res(c(Z \to X, \mathcal{N})) =
c_{top}(\mathcal{E}) \circ c(Z' \to X', \mathcal{N}')
$$
in $A^*(Z' \to X')^\wedge$.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-construction-gysin} we have
$res(c(Z \to X, \mathcal{N})) = c(Z' \to X', g^*\mathcal{N})$
and the equality follows from Lemma \ref{lemma-gysin-decompose}.
\end{proof}
\noindent
Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme
locally of finite type over $S$. Let $\mathcal{N}$ be a virtual normal
@@ -11229,7 +11267,7 @@ \section{Higher codimension gysin homomorphisms}
\begin{align*}
c(Z \to Y, \mathcal{N}') \cap c(Y \to X, \mathcal{N}'') \cap [X]
& =
c(Z \to Y, \mathcal{N}') \cap c_{top}(\mathcal{N}'') \cap [X] \\
c(Z \to Y, \mathcal{N}') \cap c_{top}(\mathcal{N}'') \cap [Y] \\
& =
c_{top}(\mathcal{N}''|_Z) \cap c(Z \to Y, \mathcal{N}') \cap [Y] \\
& =
@@ -12560,6 +12598,76 @@ \section{Gysin maps for local complete intersection morphisms}
and similarly for the chern character.
\end{remark}
\begin{lemma}
\label{lemma-compare-gysin-base-change}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
Consider a cartesian square
$$
\xymatrix{
X' \ar[d]_{f'} \ar[r]_{g'} &
X \ar[d]^f \\
Y' \ar[r]^g &
Y
}
$$
of schemes locally of finite type over $S$. Assume
\begin{enumerate}
\item both $f$ and $f'$ are local complete intersection morphisms, and
\item the gysin map exists for $f$
\end{enumerate}
Then $\mathcal{C} = \Ker(H^{-1}((g')^*\NL_{X/Y}) \to H^{-1}(\NL_{X'/Y'}))$
is a finite locally free $\mathcal{O}_{X'}$-module, the gysin map
exists for $f'$, and we have
$$
res(f^!) = c_{top}(\mathcal{C}^\vee) \circ (f')^!
$$
in $A^*(X' \to Y')$.
\end{lemma}
\begin{proof}
The fact that $\mathcal{C}$ is finite locally free follows immediately
from More on Algebra, Lemma \ref{more-algebra-lemma-base-change-lci-bis}.
Choose a factorization $f = g \circ i$ with $g : P \to Y$ smooth and $i$
an immersion. Then we can factor $f' = g' \circ i'$ where $g' : P' \to Y'$
and $i' : X' \to P'$ the base changes. Picture
$$
\xymatrix{
X' \ar[r] \ar[d] &
P' \ar[r] \ar[d] &
Y' \ar[d] \\
X \ar[r] &
P \ar[r] &
Y
}
$$
In particular, we see that the gysin map exists for $f'$. By
More on Morphisms, Lemmas \ref{more-morphisms-lemma-get-NL}
we have
$$
\NL_{X/Y} = \left( \mathcal{C}_{X/P} \to i^*\Omega_{P/Y} \right)
$$
where $\mathcal{C}_{X/P}$ is the conormal sheaf of the embedding $i$.
Similarly for the primed version. We have
$(g')^*i^*\Omega_{P/Y} = (i')^*\Omega_{P'/Y'}$ because
$\Omega_{P/Y}$ pulls back to $\Omega_{P'/Y'}$ by
Morphisms, Lemma \ref{morphisms-lemma-base-change-differentials}.
Also, recall that $(g')^*\mathcal{C}_{X/P} \to \mathcal{C}_{X'/P'}$
is surjective, see
Morphisms, Lemma \ref{morphisms-lemma-conormal-functorial-flat}.
We deduce that the sheaf $\mathcal{C}$ is canonicallly
isomorphic to the kernel of the map
$(g')^*\mathcal{C}_{X/P} \to \mathcal{C}_{X'/P'}$
of finite locally free modules. Recall that $i^!$ is defined
using $\mathcal{N} = \mathcal{C}_{Z/X}^\vee$ and similarly
for $(i')^!$. Thus we have
$$
res(i^!) = c_{top}(\mathcal{C}^\vee) \circ (i')^!
$$
in $A^*(X' \to P')$ by an application of Lemma \ref{lemma-gysin-excess}.
Since finally we have $f^! = i^! \circ g^*$,
$(f')^! = (i')^! \circ (g')^*$, and $(g')^* = res(g^*)$ we conclude.
\end{proof}
\begin{lemma}
\label{lemma-lci-gysin-product-regular}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}.

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