From f158b52323daf82795e05a2ac6fabffd1550ebca Mon Sep 17 00:00:00 2001 From: Aise Johan de Jong Date: Tue, 12 Jun 2018 16:34:33 -0400 Subject: [PATCH] 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 | 276 ++++++++++++++++++++++++++++++++++++++++++++++++++----- 1 file changed, 251 insertions(+), 25 deletions(-) diff --git a/chow.tex b/chow.tex index c212f4520..7be315978 100644 --- a/chow.tex +++ b/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} + + + + + + + + + + +