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Better handling commutativity property loc chern

Wish I had realized this a week ago. Argh!
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aisejohan committed May 20, 2019
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  1. +100 −186 chow.tex
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@@ -7875,39 +7875,28 @@ \section{A baby case of localized chern classes}

\begin{lemma}
\label{lemma-silly-commutes}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be
locally of finite type over $S$. Suppose we have closed subschemes
$X_1, X_2, X'_1, X'_2$ such that
In Lemma \ref{lemma-silly} let $f : Y \to X$ be locally of finite type
and say $c \in A^*(Y \to X)$. Then
$$
X = X_1 \cup X_2 = X'_1 \cup X'_2
c \circ P'_p(E_2) = P'_p(Lf_2^*E_2) \circ c
\quad\text{resp.}\quad
c \circ c'_p(E_2) = c'_p(Lf_2^*E_2) \circ c
$$
set theoretically. Let $E_2 \in D(\mathcal{O}_{X_2})$ and
$E'_2 \in D(\mathcal{O}_{X'_2})$ be perfect objects. Assume
\begin{enumerate}
\item chern classes of $E_2$ and $E'_2$ are defined,
\item the restrictions $E_2|_{X_1 \cap X_2}$ and $E'_2|_{X'_1 \cap X'_2}$
are isomorphic to finite locally free modules of rank $< p$ and $< p'$
sitting in cohomological degree $0$.
\end{enumerate}
Then the classes $c'_p(E_2) \in A(X_2 \to X)$ and
$c'_{p'}(E'_2) \in A(X'_2 \to X)$ of Lemma \ref{lemma-silly}
commute in the sense of Remark \ref{remark-bivariant-commute}.
in $A^*(Y_2 \to Y)$ where $f_2 : Y_2 \to X_2$ is the base change of $f$.
\end{lemma}

\begin{proof}
Let $\alpha \in A_k(X)$. We may write
$$
\alpha = \alpha_{11} + \alpha_{12} + \alpha_{21} + \alpha_{22}
\alpha = \alpha_1 + \alpha_2
$$
with $\alpha_{ij} \in A_k(X_i \cap X'_j)$; we are omitting the pushforwards
by the closed immersions $X_i \cap X'_j \to X$. Then
$c'_p(E_2) \cap \alpha = c(E_2) \cap (\alpha_{21} + \alpha_{22})$
and hence
$c'_{p'}(E'_2) \cap c'_p(E_2) \cap \alpha =
c_{p'}(E'_2) \cap c_p(E_2) \cap \alpha_{22}$
where the cap products are taking place on $X_2 \cap X'_2$.
Since chern classes commute with each other as bivariant classes
(Lemma \ref{lemma-commutative-chern-perfect}) we conclude.
with $\alpha_i \in A_k(X_i)$; we are omitting the pushforwards
by the closed immersions $X_i \to X$. The reader then checks that
$c'_p(E_2) \cap \alpha = c_p(E_2) \cap \alpha_2$,
$c \cap c'_p(E_2) \cap \alpha = c \cap c_p(E_2) \cap \alpha_2$,
$c \cap \alpha = c \cap \alpha_1 + c \cap \alpha_2$, and
$c'_p(Lf_2^*E_2) \cap c \cap \alpha = c_p(Lf_2^*E_2) \cap c \cap \alpha_2$.
We conclude by Lemma \ref{lemma-commutative-chern-perfect}.
\end{proof}

\begin{lemma}
@@ -8127,7 +8116,41 @@ \section{Gysin at infinity}
and we get the desired result.
\end{proof}

\begin{lemma}
\label{lemma-gysin-at-infty-commutes}
In Lemma \ref{lemma-gysin-at-infty} let $f : Y \to X$ be a morphism
locally of finite type and $c \in A^*(Y \to X)$. Then $C \circ c = c \circ C$
in $A^*(W_\infty \times_X Y)$.
\end{lemma}

\begin{proof}
Consider the commutative diagram
$$
\xymatrix{
W_\infty \times_X Y \ar@{=}[r] &
W_{Y, \infty} \ar[r]_{i_{Y, \infty}} \ar[d] &
W_Y \ar[r]_{b_Y} \ar[d] &
\mathbf{P}^1_Y \ar[r]_{p_Y} \ar[d] &
Y \ar[d]^f \\
& W_\infty \ar[r]^{i_\infty} &
W \ar[r]^b &
\mathbf{P}^1_X \ar[r]^p &
X
}
$$
with cartesian squares. For an elemnent $\alpha \in A_k(X)$
choose $\beta \in A_{k + 1}(W)$ whose restriction to $b^{-1}(\mathbf{A}^1_X)$
is the flat pullback of $\alpha$. Then $c \cap \beta$ is a class
in $A_*(W_Y)$ whose restriction to $b_Y^{-1}(\mathbf{A}^1_Y)$
is the flat pullback of $c \cap \alpha$. Next, we have
$$
i_{Y, \infty}^*(c \cap \beta) = c \cap i_\infty^*\beta
$$
because $c$ is a bivariant class. This exactly says that
$C \cap c \cap \alpha = c \cap C \cap \alpha$. The same argument
works after any base change by $X' \to X$ locally of finite type.
This proves the lemma.
\end{proof}



@@ -8322,43 +8345,30 @@ \section{Preparation for localized chern classes}

\begin{lemma}
\label{lemma-homomorphism-commute}
In Lemma \ref{lemma-localized-chern-pre} assume $Q|_T$ is isomorphic
to a finite locally free $\mathcal{O}_T$-module of rank $< p$.
Assume we have another perfect object $Q' \in D(\mathcal{O}_W)$
whose chern classes are defined with $Q'|_T$ is isomorphic to a
finite locally free $\mathcal{O}_T$-module of rank $< p'$ placed
in cohomological degree $0$. Then $c'_p(Q) \in A^p(Z \to X)$
and $c'_{p'}(Q') \in A^{p'}(Z \to X)$ commute in the sense of
Remark \ref{remark-bivariant-commute}.
In Lemma \ref{lemma-localized-chern-pre} let $Y \to X$ be a morphism
locally of finite type and let $c \in A^*(Y \to X)$ be a bivariant class.
Then
$$
P'_p(Q) \circ c = c \circ P'_p(Q)
\quad\text{resp.}\quad
c'_p(Q) \circ c = c \circ c'_p(Q)
$$
in $A^*(Y \times_X Z \to X)$.
\end{lemma}

\begin{proof}
Let $E \subset W_\infty$ be the inverse image of $Z$.
Denote $c'_p(Q|_E)$ and $c'_{p'}(Q'|_E)$ the classes
constructed in Lemma \ref{lemma-silly}. We have
\begin{align*}
res(c'_p(Q)) \circ c'_{p'}(Q')
& =
c'_p(Q) \circ (Z \to X)_* \circ c'_{p'}(Q') \\
& =
(E \to Z)_* \circ c'_p(Q|_E) \circ C \circ (Z \to X)_* \circ c'_{p'}(Q') \\
& =
(E \to Z)_* \circ c'_p(Q|_E) \circ c_{p'}(Q'|_{W_\infty}) \circ C \\
& =
(E \to Z)_* \circ c'_p(Q|_E) \circ
(E \to W_\infty)_* \circ c'_{p'}(Q'|_E) \circ C \\
& =
(E \to Z)_* \circ res(c'_p(Q|_E)) \circ c'_{p'}(Q'|_E) \circ C
\end{align*}
The first equality holds by Remark \ref{remark-res-push}.
The second equality is the definition of $c'_p(Q)$.
The third equality is Lemma \ref{lemma-homomorphism}.
The fourth equality is Lemma \ref{lemma-silly-silly}.
The fifth equality is Remark \ref{remark-res-push} once again
but this time using $E \subset W_\infty$.
By Lemma \ref{lemma-silly-commutes} the order in the product
$res(c'_p(Q|_E)) \circ c'_{p'}(Q'|_E)$ may be switched and
we conclude.
Recall that $P'_p(Q) = (E \to Z)_* \circ P'_p(Q|_E) \circ C$,
resp.\ $c'_p(Q) = (E \to Z)_* \circ c'_p(Q|_E) \circ C$
where $C$ is as in Lemma \ref{lemma-gysin-at-infty} and
$P'_p(Q|_E)$, resp.\ $c'_p(Q|_E)$ are as in
Lemma \ref{lemma-silly}.
By Lemma \ref{lemma-gysin-at-infty-commutes}
we see that $C$ commutes with $c$
and by Lemma \ref{lemma-silly-commutes} we see that
$P'_p(Q|_E)$, resp.\ $c'_p(Q|_E)$ commutes with $c$.
Since $c$ is a bivariant class it commutes with proper
pushforward by $E \to Z$ by definition. This finishes the proof.
\end{proof}

\begin{lemma}
@@ -8907,146 +8917,50 @@ \section{Properties of localized chern classes}

\begin{lemma}
\label{lemma-loc-chern-classes-commute}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme
locally of finite type over $S$. Let $Z_i \subset X$, $i = 1, 2$
be closed subschemes. Let $E_i \in D(\mathcal{O}_X)$, $i = 1, 2$
be a perfect objects whose chern classes are defined.
Assume the restriction $E_i|_{X \setminus Z_i}$ is isomorphic to a
finite locally free $\mathcal{O}_{X \setminus Z_i}$-module of rank $< p_i$
sitting in cohomological degree $0$.
Then $c_{p_1}(Z_1 \to X, E_1)$ and $c_{p_2}(Z_2 \to X, E_2)$
commute in the sense of Remark \ref{remark-bivariant-commute}.
In the situation of Definition \ref{definition-localized-chern}
assume $P_p(Z \to X, E)$, resp.\ $c_p(Z \to X, E)$ is defined.
Let $Y \to X$ be locally of finite type and $c \in A^*(Y \to X)$.
Then
$$
P_p(Z \to X, E) \circ c = c \circ P_p(Z \to X, E),
$$
respectively
$$
c_p(Z \to X, E) \circ c = c \circ c_p(Z \to X, E)
$$
in $A^*(Y \times_X Z \to X)$.
\end{lemma}

\begin{proof}
The difficulty in this proof is that we don't assume that $Z_1 = Z_2$
so we can't reduce this lemma to Lemma \ref{lemma-homomorphism-commute}.
We'll actually have to use something about the precise complexes we
use in the construction of localized chern classes.

\medskip\noindent
Our assumptions say $E_i$
is represented to a bounded complex $\mathcal{E}_i^\bullet$
of finite locally free $\mathcal{O}_X$-modules. For $i = 1, 2$ let
This follows from Lemma \ref{lemma-homomorphism-commute}.
Namely, our assumptions say $E$
is represented to a bounded complex $\mathcal{E}^\bullet$
of finite locally free $\mathcal{O}_X$-modules. Let
$$
b_i : W_i \to \mathbf{P}^1_X
b : W \to \mathbf{P}^1_X
\quad\text{and}\quad
\mathcal{Q}_i^\bullet
\mathcal{Q}^\bullet
$$
be the blowing up and complex of $\mathcal{O}_{W_i}$-modules constructed in
be the blowing up and complex of $\mathcal{O}_W$-modules constructed in
More on Flatness, Section \ref{flat-section-blowup-complexes-III}.
Let $T_i \subset W_{i, \infty}$ be the closed subscheme whose existence is
Let $T \subset W_\infty$ be the closed subscheme whose existence is
averted in More on Flatness, Lemma \ref{flat-lemma-graph-construction}.
Let $T'_i \subset T_i$ be the open and closed subscheme such that
$\mathcal{Q}_i^\bullet|_{T'_i}$ is isomorphic to a finite locally free
sheaf of rank $< p_i$ in degree $0$. By definition
Let $T' \subset T$ be the open and closed subscheme such that
$\mathcal{Q}_i^\bullet|_{T'_i}$ is zero, resp.\ isomorphic to a
finite locally free sheaf of rank $< p$ placed in degree $0$. By definition
$$
c_p(Z \to X, E_i) = c'_p(\mathcal{Q}_i^\bullet)
c_p(Z \to X, E) = c'_p(\mathcal{Q}^\bullet)
$$
as bivariant operations (and not just on cycles over $X$)
where the right hand side is the bivariant class constructed in
Lemma \ref{lemma-localized-chern-pre} using
$W_i, b_i, \mathcal{Q}_i^\bullet, T'_i$.

\medskip\noindent
By Divisors, Lemma \ref{divisors-lemma-blowing-up-two-ideals} we
can choose a commutative diagram
$$
\xymatrix{
W \ar[d]^{g_1} \ar[rd]^b \ar[r]_{g_2} & W_2 \ar[d]^{b_2} \\
W_1 \ar[r]^{b_1} & \mathbf{P}^1_X
}
$$
where all morphisms are blowing ups which are isomorphisms over
$\mathbf{A}^1_X$. By Lemma \ref{lemma-localized-chern-pre-independent}
we may use $W$, $b = b_i \circ g_i$, $Q_i = g_i^*\mathcal{Q}_i^\bullet$, and
$g_i^{-1}(T'_i)$ to construct $c_{p_i}(Z \to X, E_i) = c'_{p_i}(Q_i)$.
By More on Flatness, Lemma \ref{flat-lemma-complex-and-divisor-eta-pull}
applied to the morphisms $g_i : W \to W_i$ we find that
Lemma \ref{lemma-localized-chern-pre} using $W, b, \mathcal{Q}^\bullet, T'$.
By Lemma \ref{lemma-homomorphism-commute} we have
$$
Q_i = g_i^*\mathcal{Q}_i^\bullet =
\eta_\mathcal{I}b^*p^*\mathcal{E}_i^\bullet
$$
where $\mathcal{I}$ is the invertible ideal sheaf of the effective
Cartier divisor $W_\infty$ and $p :\mathbf{P}^1_X \to X$
is the projection morphism.

\medskip\noindent
Set $W^i = b^{-1}(p^{-1}(Z_i)) \subset W$. Picture
$$
\xymatrix{
W^1 \ar[d]_{b^1} \ar[r] & W \ar[d]^b & W^2 \ar[l] \ar[d]^{b^2} \\
\mathbf{P}^1_{Z_1} \ar[r] & \mathbf{P}^1_X & \mathbf{P}^1_{Z_2} \ar[l]
}
$$
Since $\mathcal{E}_i^\bullet|_{X \setminus Z_i}$ is isomorphic to a
finite locally free $\mathcal{O}_{X \setminus Z_i}$-module of rank $< p_i$
sitting in cohomological degree $0$, we see that
$b^*p^*\mathcal{E}_i^\bullet|_{W \setminus W^i}$
is isomorphic to a finite locally free
$\mathcal{O}_{W \setminus W^i}$-module
of rank $< p_i$ sitting in cohomological degree $0$. Hence,
for example by More on Flatness, Lemma \ref{flat-lemma-eta-first-property},
we see $Q_i|_{W \setminus W^i}$ is isomorphic a finite locally free
$\mathcal{O}_{W \setminus W^i}$-module
of rank $< p_i$ sitting in cohomological degree $0$.

\medskip\noindent
Let $\alpha \in A_k(X)$. We are going to compute
$c_{p_2}(Z_2 \to X, E_2) \cap c_{p_1}(Z_1 \to X, E_1) \cap \alpha$
in $A_{k - p_1 - p_2}(X)$.
Choose $\beta \in A_{k + 1}(W)$ whose restriction
to $b^{-1}(\mathbf{A}^1_X)$ is the flat pullback of $\alpha$. Since
$W^1_\infty$ is the inverse image of $Z_1$ under the morphism
$W_\infty \to X$ we have the first equality of
\begin{align*}
c_{p_1}(Z_1 \to X, E_1) \cap \alpha
& =
(W^1_\infty \to Z_1)_*(c'_{p_1}(Q_1|_{W^1_\infty}) \cap i_\infty^*\beta) \\
& =
(W^1_\infty \to Z_1)_*(
c_{p_1}(W^1_\infty \to W_\infty, Q_1|_{W_\infty}) \cap i_\infty^*\beta) \\
& =
(W^1_\infty \to Z_1)_* (i^1_\infty)^*
(c_{p_1}(W^1 \to W, Q_1) \cap \beta)
\end{align*}
The second equality is Lemma \ref{lemma-loc-chern-agree}.
The third equality is the fact that $c_{p_1}(W^1 \to W, Q_1)$
is a bivariant class whose formation commutes with base change and
$i^1_\infty : W^1_\infty \to W^1$ is the base change of $i_\infty$.
Note that the proper morphism
$$
b^1 : W^1 \longrightarrow \mathbf{P}^1_{Z_1}
$$
and the restriction $Q_2|_{W^1}$ is the pair which defines the restriction of
$c_{p_2}(Z_2 \to X, E_2) = c'_{p_2}(Q_2)$ to $Z_1$. Denote
$C_1 \in A^0(W^1_\infty \to Z_1)$ the class of Lemma \ref{lemma-gysin-at-infty}.
Then we have to compute
$$
((W^1 \cap W^2)_\infty \to Z_1 \cap Z_2)_*\Big(
c'_{p_2}(Q_2|_{(W^1 \cap W^2)_\infty}) \cap C_1 \cap
(W^1_\infty \to Z_1)_* (i^1_\infty)^*
(c_{p_1}(W^1 \to W, Q_1) \cap \beta)\Big)
$$
with apologies for the horrible notation.
By Lemma \ref{lemma-homomorphism-pre} we have
$C_1 \circ (W^1_\infty \to Z_1)_* \circ (i^1_\infty)^* = (i^1_\infty)^*$
hence this simplifies to
$$
((W^1 \cap W^2)_\infty \to Z_1 \cap Z_2)_*\Big(
c'_{p_2}(Q_2|_{(W^1 \cap W^2)_\infty}) \cap (i^1_\infty)^*
(c_{p_1}(W^1 \to W, Q_1) \cap \beta)\Big)
$$
which in turn is equal to
$$
((W^1 \cap W^2)_\infty \to Z_1 \cap Z_2)_*(
c'_{p_2}(Q_2|_{(W^1 \cap W^2)_\infty}) \cap
c'_{p_1}(Q_1|_{W^1_\infty}) \cap i_\infty^*\beta)
P'_p(\mathcal{Q}^\bullet) \circ c = c \circ P'_p(\mathcal{Q}^\bullet)
\quad\text{resp.}\quad
c'_p(\mathcal{Q}^\bullet) \circ c = c \circ c'_p(\mathcal{Q}^\bullet)
$$
by reusing some of the equalities we used above (in the opposite direction).
Since $c'_{p_2}(Q_2|_{(W^1 \cap W^2)_\infty})$ is the
restriction of $c'_{p_2}(Q_2|_{W^2_\infty})$ we find that
this is symmetric in the indices by Lemma \ref{lemma-silly-commutes}
which concludes the proof.
in $A^*(Y \times_X Z \to X)$ and we conclude.
\end{proof}

\begin{remark}
@@ -9841,7 +9755,7 @@ \section{Higher codimension gysin homomorphisms}
c(Z \to X, \mathcal{N}) =
c_{top}(\mathcal{E}) \circ i^*
$$
where $i^*$ is the gysing homomorphism associated to $i : Z \to X$
where $i^*$ is the gysin 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}.

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