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Chern classes of complexes

For the moment we can only prove that the result is well defined, i.e.,
independent of the global resolution.
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aisejohan committed Jun 12, 2018
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276 chow.tex
@@ -4841,7 +4841,7 @@ \section{Bivariant intersection theory}
\begin{enumerate}
\item if $Y'' \to Y'$ is a proper, then
$c \cap (Y'' \to Y')_*\alpha'' = (X'' \to X')_*(c \cap \alpha'')$
for all $\alpha''$ on $Y''$,
for all $\alpha''$ on $Y''$ where $X'' = Y'' \times_Y X$,
\item if $Y'' \to Y'$ is flat locally of finite type of
fixed relative dimension, then
$c \cap (Y'' \to Y')^*\alpha' = (X'' \to X')^*(c \cap \alpha')$
@@ -4957,8 +4957,8 @@ \section{Bivariant intersection theory}
Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of
schemes locally of finite tyoe over $S$.
Let $c \in A^p(X \to Z)$ and assume $f$ is proper.
Then the rule that to $X' \to X$ assignes
$\alpha \longmapsto f_*(c \cap \alpha)$
Then the rule that to $Z' \to Z$ assignes
$\alpha \longmapsto f'_*(c \cap \alpha)$
is a bivariant class of degree $p$.
\end{lemma}
@@ -4984,7 +4984,7 @@ \section{Bivariant intersection theory}
which assigns to every locally of finite type morphism $Y' \to Y$
and every $k$ a map
$$
c \cap - : Z_k(X') \longrightarrow A_{k - p}(Y')
c \cap - : Z_k(Y') \longrightarrow A_{k - p}(X')
$$
where $Y' = X' \times_X Y$, satisfying condition (3) of
Definition \ref{definition-bivariant-class}
@@ -5063,26 +5063,39 @@ \section{Bivariant intersection theory}
\begin{lemma}
\label{lemma-bivariant-zero}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
Let $X$ be locally of finite type over $S$. Let $c \in A^p(X)$.
Then $c$ is zero if and only if $c \cap [Y] = 0$ in $A_*(Y)$
for every integral scheme $Y$ locally of finite type over $X$.
Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$.
Let $c \in A^p(X \to Y)$. For $Y'' \to Y' \to Y$ set
$X'' = Y'' \times_Y X$ and $X' = Y' \times_Y X$.
The following are equivalent
\begin{enumerate}
\item $c$ is zero,
\item $c \cap [Y'] = 0$ in $A_*(X')$ for every integral scheme $Y'$
locally of finite type over $Y$, and
\item for every integral scheme $Y'$ locally of finite type over $Y$,
there exists a proper birational morphism $Y'' \to Y'$ such that
$c \cap [Y''] = 0$ in $A_*(X'')$.
\end{enumerate}
\end{lemma}
\begin{proof}
The if direction is clear. For the converse, assume that $c \cap [Y] = 0$ in
$A_*(Y)$ for every integral scheme $Y$ locally of finite type over $X$.
Let $X' \to X$ be locally of finite type. Let $\alpha \in A_k(X')$.
Write $\alpha = \sum n_i [Y_i]$ with $Y_i \subset X'$ a locally finite
The implications (1) $\Rightarrow$ (2) $\Rightarrow$ (3) are clear.
Assumption (3) imlpies (2) because $(Y'' \to Y')_*[Y''] = [Y']$
and hence $c \cap [Y'] = (X'' \to X')_*(c \cap [Y''])$ as $c$
is a bivariant class. Assume (2).
Let $Y' \to Y$ be locally of finite type. Let $\alpha \in A_k(Y')$.
Write $\alpha = \sum n_i [Y'_i]$ with $Y'_i \subset Y'$ a locally finite
collection of integral closed subschemes of $\delta$-dimension $k$.
Then we see that $\alpha$ is pushforward of the cycle
$\alpha' = \sum n_i[Y_i]$ on $X'' = \coprod Y_i$ under the
proper morphism $X'' \to X'$. By the properties of bivariant
$\alpha' = \sum n_i[Y'_i]$ on $Y'' = \coprod Y'_i$ under the
proper morphism $Y'' \to Y'$. By the properties of bivariant
classes it suffices to prove that $c \cap \alpha' = 0$ in $A_{k - p}(X'')$.
We have $A_{k - p}(X'') = \prod A_{k - p}(Y_i)$ as follows immediately
from the definitions. The projection maps $A_{k - p}(X'') \to A_{k - p}(Y_i)$
are given by flat pullback. Since capping with $c$ commutes with
flat pullback, we see that it suffices to show that $c \cap [Y_i]$
is zero in $A_{k - p}(Y_i)$ which is true by assumption.
We have $A_{k - p}(X'') = \prod A_{k - p}(X'_i)$ where
$X'_i = Y'_i \times_Y X$. This follows immediately
from the definitions. The projection maps
$A_{k - p}(X'') \to A_{k - p}(X'_i)$ are given by flat pullback.
Since capping with $c$ commutes with
flat pullback, we see that it suffices to show that $c \cap [Y'_i]$
is zero in $A_{k - p}(X'_i)$ which is true by assumption.
\end{proof}
@@ -5480,12 +5493,21 @@ \section{Intersecting with chern classes}
\label{section-intersecting-chern-classes}
\noindent
In this section we study the operation of capping with chern
classes of vector bundles. Our definition follows the familiar
In this section we define chern classes of vector bundles on $X$ as
bivariant classes on $X$, see Lemma \ref{lemma-cap-cp-bivariant}
and the discussion following this lemma. Our construction follows the familiar
pattern of first defining the operation on prime cycles and then
summing, but in Lemma \ref{lemma-determine-intersections} we show
summing. In Lemma \ref{lemma-determine-intersections} we show
that the result is determined by the usual formula on the associated
projective bundle.
projective bundle. Next, we show that capping with chern classes
passes through rational equivalence, commutes with proper pushforward,
commutes with flat pullback, and commutes with the gysin maps for
inclusions of effective Cartier divisors. These lemmas could have been
avoided by directly using the characterization in
Lemma \ref{lemma-determine-intersections} and using
Lemma \ref{lemma-push-proper-bivariant}; the reader who wishes to
see this worked out should consult
Chow Groups of Spaces, Lemma \ref{spaces-chow-lemma-segre-classes}.
\begin{definition}
\label{definition-cap-chern-classes}
@@ -5850,6 +5872,32 @@ \section{Intersecting with chern classes}
This proves the lemma.
\end{proof}
\begin{remark}
\label{remark-extend-to-finite-locally-free}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
Let $X$ be locally of finite type over $S$.
Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module.
If the rank of $\mathcal{E}$ is not constant then we can
still define the chern classes of $\mathcal{E}$. Namely, in this
case we can write
$$
X = X_0 \amalg X_1 \amalg X_2 \amalg \ldots
$$
where $X_r \subset X$ is the open and closed subspace where
the rank of $\mathcal{E}$ is $r$. If $X' \to X$ is a morphism
which is locally of finite type, then we obtain by
pullback a corresponding decomposition of $X'$ and we find that
$$
A_*(X') = \prod\nolimits_{r \geq 0} A_*(X'_r)
$$
by our definitions. Then we simply define $c_i(\mathcal{E})$
to be the bivariant class which preserves these direct
product decompositions and acts by the already defined
operations $c_i(\mathcal{E}|_{X_r}) \cap -$
on the factors. Observe that in this setting it may happen
that $c_i(\mathcal{E})$ is nonzero for infinitely many $i$.
\end{remark}
@@ -6322,7 +6370,8 @@ \section{The splitting principle}
\section{Chern classes and tensor product}
\section{The Chern character and tensor products}
\label{section-chern-classes-tensor}
\noindent
@@ -6332,8 +6381,7 @@ \section{Chern classes and tensor product}
ch({\mathcal E}) = \sum\nolimits_{i=1}^r e^{x_i}
$$
if the $x_i$ are the chern roots of ${\mathcal E}$. Writing this in
terms of chern classes $c_i = c_i(\mathcal{E})$
we see that
terms of chern classes $c_i = c_i(\mathcal{E})$ we see that
$$
ch(\mathcal{E}) =
r
@@ -6370,6 +6418,184 @@ \section{Chern classes and tensor product}
This follows directly from the discussion of the chern roots
of the tensor product in the previous section.
\begin{remark}
\label{remark-extend-chern-character-to-finite-locally-free}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
Let $X$ be locally of finite type over $S$.
Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module.
If the rank of $\mathcal{E}$ is not constant then we can
still define the Chern character $ch(\mathcal{E})$
of $\mathcal{E}$, exactly as in
Remark \ref{remark-extend-to-finite-locally-free}.
It is still the case that $ch_i(\mathcal{E})$
is in $A^i(X) \otimes \mathbf{Q}$ with denominator at worst $i!$.
\end{remark}
\section{Chern classes and the derived category}
\label{section-pre-derived}
\noindent
In this section we define the total chern class of an object
of the derived category which may be represented globally
by a finite complex of finite locally free modules.
\medskip\noindent
Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
Let $X$ be locally of finite type over $S$. Let
$$
\mathcal{E}^a \to \mathcal{E}^{a + 1} \to \ldots \to \mathcal{E}^b
$$
be a finite complex of finite locally free $\mathcal{O}_X$-modules.
Then we define the {\it total chern class of the complex} by the formula
$$
c(\mathcal{E}^\bullet) = \prod\nolimits_{p = a, \ldots, b}
c(\mathcal{E}^p)^{(-1)^p}
$$
in $A^*(X)$. Here the inverse is the formal inverse, so
$$
(1 + c_1 + c_2 + c_3 + \ldots)^{-1} =
1 - c_1 + c_1^2 - c_2 - c_1^3 + 2c_1 c_2 - c_3 + \ldots
$$
We similarly define the {\it Chern character of the complex} by
the formula
$$
ch(\mathcal{E}^\bullet) = \sum\nolimits_{p = a, \ldots, b}
(-1)^p ch(\mathcal{E}^p)
$$
in $A^*(X) \otimes \mathbf{Q}$.
Let us prove that $c(\mathcal{E}^\bullet)$ only depends
on the image of the complex
in the derived category.
\begin{lemma}
\label{lemma-pre-derived-chern-class}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
Let $X$ be locally of finite type over $S$. Let $E \in D(\mathcal{O}_X)$
be an object such that there exists a finite complex $\mathcal{E}^\bullet$
of finite locally free $\mathcal{O}_X$-modules representing $E$.
Then $c(\mathcal{E}^\bullet) \in A^*(X)$ is independent of the
choice of the complex. Similarly for $ch(\mathcal{E}^\bullet)$.
\end{lemma}
\begin{proof}
Suppose we have a second finite complex $\mathcal{F}^\bullet$
of finite locally free $\mathcal{O}_X$-modules representing $E$.
Choose $a \leq b$ such that $\mathcal{F}^p$ and $\mathcal{E}^p$
are zero for $p \not \in [a, b]$. We will prove the lemma
by induction on $b - a$. If $b - a = 0$, then we have
$\mathcal{F}^a \cong \mathcal{E}^a \cong E$ and the result is clear.
\medskip\noindent
Induction step. Assume $b > a$. Let $g : Y \to X$ be a morphism
locally of finite type with $Y$ integral.
By Lemma \ref{lemma-bivariant-zero} it suffices to show that
with $c(g^*\mathcal{E}^\bullet) \cap [Y]$ is the same as
$c(g^*\mathcal{F}^\bullet) \cap [Y]$ and it even suffices to prove
this after replacing $Y$ by an integral scheme proper and birational
over $Y$. By
More on Flatness, Lemma \ref{flat-lemma-blowup-complex-integral}
we may assume that $H^b(Lg^*E)$ is perfect of tor dimension $\leq 1$.
This reduces us to the case discussed in the next paragraph.
\medskip\noindent
Assume $X$ is integral and $H^b(E)$ is a perfect $\mathcal{O}_X$-module
of tor dimension $\leq 1$. Let
$$
\mathcal{G} = \Ker(\mathcal{E}^b \oplus \mathcal{F}^b \to H^b(E))
$$
Since $H^b(E)$ has tor dimension $\leq 1$ we see that
$\mathcal{G}$ is finite locally free. Then there is a commutative diagram
$$
\xymatrix{
\mathcal{G}[-b] \ar[r]_\alpha \ar[d]^\beta & \mathcal{E}^\bullet \ar[d] \\
\mathcal{F}^\bullet \ar[r] & E
}
$$
in $D(\mathcal{O}_X)$.
(Warning: you have to choose the negative of the canonical map for one of
the arrows to make this diagram commute.)
Choose a distinguished triangle
$$
\mathcal{G}[-b] \to E \to E' \to \mathcal{G}[-b + 1]
$$
in $D(\mathcal{O}_X)$. On the other hand, the cone on
$\alpha : \mathcal{G}[-b] \to \mathcal{E}^\bullet$
gives a distinguished triangle
$$
\mathcal{G}[-b] \to
\mathcal{E}^\bullet \to C(\alpha) \to
\mathcal{G}[-b + 1]
$$
and similarly for $\mathcal{F}^\bullet$ and $\beta$.
Since $\mathcal{G} \to \mathcal{E}^b$ is surjective,
it follows that $C(\alpha)$ has vanishing cohomogy
in degree $b$ and hence
$$
\tau_{\leq b - 1}C(\alpha) \longrightarrow C(\alpha)
$$
is an isomorphism in $D(\mathcal{O}_X)$. On the other hand, the displayed
arrow determines an isomorphism of complexes except in degrees
$b - 1$ and $b$ where we have
$$
\left(\tau_{\leq b - 1}C(\alpha)\right)^{b - 1} =
\Ker\left(C(\alpha)^{b - 1} \to C(\alpha)^b\right)
\quad\text{and}\quad
\left(\tau_{\leq b - 1}C(\alpha)\right)^b = 0
$$
Since $C(\alpha)^{b - 1} \to C(\alpha)^b$ is a surjection of finite
locally free $\mathcal{O}_X$-modules, we conclude from multiplicativity
of total chern classes (Lemma \ref{lemma-additivity-chern-classes})
$$
c(\tau_{\leq b - 1}C(\alpha)) = c(C(\alpha))
$$
and similarly for $C(\beta)$. By the axioms of a triangulated category we
obtain an isomorphism $C(\alpha) \to E'$ in $D(\mathcal{O}_X)$
and similarly of $\mathcal{F}^\bullet$. By induction hypothesis we obtain
$$
c(\tau_{\leq b - 1}C(\alpha)) \cap [X] =
c(\tau_{\leq b - 1}C(\beta)) \cap [X]
$$
We conclude that
$$
c(\tau_{\leq b - 1}C(\alpha)) \cap [X] =
c(C(\alpha)) \cap [X] =
c(\mathcal{E}^\bullet) c(\mathcal{G})^{(-1)^{b - 1}} \cap [X]
$$
and similarly for $\mathcal{F}^\bullet$. The second equality follows
because the terms of $C(\alpha)$ are identical to the terms of
the complex $\mathcal{E}^\bullet$, except in degree $b - 1$
we've added $\mathcal{G}$ (plus we use
Lemma \ref{lemma-additivity-chern-classes} again).
We conclude that
$$
c(\mathcal{E}^\bullet) c(\mathcal{G})^{(-1)^{b - 1}} \cap [X] =
c(\mathcal{F}^\bullet) c(\mathcal{G})^{(-1)^{b - 1}} \cap [X]
$$
and we win since multiplying by a total chern class is an
invertible operation.
\end{proof}

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