Chern classes of complexes

For the moment we can only prove that the result is well defined, i.e.,
independent of the global resolution.

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}
+
+
+
+
+
+
+
+
+
+
+