42.38 Intersecting with Chern classes
In this section we define Chern classes of vector bundles on $X$ as bivariant classes on $X$, see Lemma 42.38.7 and the discussion following this lemma. Our construction follows the familiar pattern of first defining the operation on prime cycles and then summing. In Lemma 42.38.2 we show that the result is determined by the usual formula on the associated 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 42.38.2 and using Lemma 42.33.4; the reader who wishes to see this worked out should consult Chow Groups of Spaces, Lemma 82.28.1.
Definition 42.38.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. We define, for every integer $k$ and any $0 \leq j \leq r$, an operation
\[ c_ j(\mathcal{E}) \cap - : Z_ k(X) \to \mathop{\mathrm{CH}}\nolimits _{k - j}(X) \]
called intersection with the $j$th Chern class of $\mathcal{E}$.
Given an integral closed subscheme $i : W \to X$ of $\delta $-dimension $k$ we define
\[ c_ j(\mathcal{E}) \cap [W] = i_*(c_ j({i^*\mathcal{E}}) \cap [W]) \in \mathop{\mathrm{CH}}\nolimits _{k - j}(X) \]
where $c_ j({i^*\mathcal{E}}) \cap [W]$ is as defined in Definition 42.37.1.
For a general $k$-cycle $\alpha = \sum n_ i [W_ i]$ we set
\[ c_ j(\mathcal{E}) \cap \alpha = \sum n_ i c_ j(\mathcal{E}) \cap [W_ i] \]
If $\mathcal{E}$ has rank $1$ then this agrees with our previous definition (Definition 42.25.1) by Lemma 42.37.2.
Lemma 42.38.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $(\pi : P \to X, \mathcal{O}_ P(1))$ be the projective bundle associated to $\mathcal{E}$. For $\alpha \in Z_ k(X)$ the elements $c_ j(\mathcal{E}) \cap \alpha $ are the unique elements $\alpha _ j$ of $\mathop{\mathrm{CH}}\nolimits _{k - j}(X)$ such that $\alpha _0 = \alpha $ and
\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_ P(1))^ i \cap \pi ^*(\alpha _{r - i}) = 0 \]
holds in the Chow group of $P$.
Proof.
The uniqueness of $\alpha _0, \ldots , \alpha _ r$ such that $\alpha _0 = \alpha $ and such that the displayed equation holds follows from the projective space bundle formula Lemma 42.36.2. The identity holds by definition for $\alpha = [W]$ where $W$ is an integral closed subscheme of $X$. For a general $k$-cycle $\alpha $ on $X$ write $\alpha = \sum n_ a[W_ a]$ with $n_ a \not= 0$, and $i_ a : W_ a \to X$ pairwise distinct integral closed subschemes. Then the family $\{ W_ a\} $ is locally finite on $X$. Set $P_ a = \pi ^{-1}(W_ a) = \mathbf{P}(\mathcal{E}|_{W_ a})$. Denote $i'_ a : P_ a \to P$ the corresponding closed immersions. Consider the fibre product diagram
\[ \xymatrix{ P' \ar@{=}[r] \ar[d]_{\pi '} & \coprod P_ a \ar[d]_{\coprod \pi _ a} \ar[r]_{\coprod i'_ a} & P \ar[d]^\pi \\ X' \ar@{=}[r] & \coprod W_ a \ar[r]^{\coprod i_ a} & X } \]
The morphism $p : X' \to X$ is proper. Moreover $\pi ' : P' \to X'$ together with the invertible sheaf $\mathcal{O}_{P'}(1) = \coprod \mathcal{O}_{P_ a}(1)$ which is also the pullback of $\mathcal{O}_ P(1)$ is the projective bundle associated to $\mathcal{E}' = p^*\mathcal{E}$. By definition
\[ c_ j(\mathcal{E}) \cap [\alpha ] = \sum i_{a, *}(c_ j(\mathcal{E}|_{W_ a}) \cap [W_ a]). \]
Write $\beta _{a, j} = c_ j(\mathcal{E}|_{W_ a}) \cap [W_ a]$ which is an element of $\mathop{\mathrm{CH}}\nolimits _{k - j}(W_ a)$. We have
\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_{P_ a}(1))^ i \cap \pi _ a^*(\beta _{a, r - i}) = 0 \]
for each $a$ by definition. Thus clearly we have
\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_{P'}(1))^ i \cap (\pi ')^*(\beta _{r - i}) = 0 \]
with $\beta _ j = \sum n_ a\beta _{a, j} \in \mathop{\mathrm{CH}}\nolimits _{k - j}(X')$. Denote $p' : P' \to P$ the morphism $\coprod i'_ a$. We have $\pi ^*p_*\beta _ j = p'_*(\pi ')^*\beta _ j$ by Lemma 42.15.1. By the projection formula of Lemma 42.26.4 we conclude that
\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_ P(1))^ i \cap \pi ^*(p_*\beta _ j) = 0 \]
Since $p_*\beta _ j$ is a representative of $c_ j(\mathcal{E}) \cap \alpha $ we win.
$\square$
We will consistently use this characterization of Chern classes to prove many more properties.
Lemma 42.38.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. If $\alpha \sim _{rat} \beta $ are rationally equivalent $k$-cycles on $X$ then $c_ j(\mathcal{E}) \cap \alpha = c_ j(\mathcal{E}) \cap \beta $ in $\mathop{\mathrm{CH}}\nolimits _{k - j}(X)$.
Proof.
By Lemma 42.38.2 the elements $\alpha _ j = c_ j(\mathcal{E}) \cap \alpha $, $j \geq 1$ and $\beta _ j = c_ j(\mathcal{E}) \cap \beta $, $j \geq 1$ are uniquely determined by the same equation in the chow group of the projective bundle associated to $\mathcal{E}$. (This of course relies on the fact that flat pullback is compatible with rational equivalence, see Lemma 42.20.2.) Hence they are equal.
$\square$
In other words capping with Chern classes of finite locally free sheaves factors through rational equivalence to give maps
\[ c_ j(\mathcal{E}) \cap - : \mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _{k - j}(X). \]
Our next task is to show that Chern classes are bivariant classes, see Definition 42.33.1.
Lemma 42.38.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $p : X \to Y$ be a proper morphism. Let $\alpha $ be a $k$-cycle on $X$. Let $\mathcal{E}$ be a finite locally free sheaf on $Y$. Then
\[ p_*(c_ j(p^*\mathcal{E}) \cap \alpha ) = c_ j(\mathcal{E}) \cap p_*\alpha \]
Proof.
Let $(\pi : P \to Y, \mathcal{O}_ P(1))$ be the projective bundle associated to $\mathcal{E}$. Then $P_ X = X \times _ Y P$ is the projective bundle associated to $p^*\mathcal{E}$ and $\mathcal{O}_{P_ X}(1)$ is the pullback of $\mathcal{O}_ P(1)$. Write $\alpha _ j = c_ j(p^*\mathcal{E}) \cap \alpha $, so $\alpha _0 = \alpha $. By Lemma 42.38.2 we have
\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_ P(1))^ i \cap \pi _ X^*(\alpha _{r - i}) = 0 \]
in the chow group of $P_ X$. Consider the fibre product diagram
\[ \xymatrix{ P_ X \ar[r]_-{p'} \ar[d]_{\pi _ X} & P \ar[d]^\pi \\ X \ar[r]^ p & Y } \]
Apply proper pushforward $p'_*$ (Lemma 42.20.3) to the displayed equality above. Using Lemmas 42.26.4 and 42.15.1 we obtain
\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_ P(1))^ i \cap \pi ^*(p_*\alpha _{r - i}) = 0 \]
in the chow group of $P$. By the characterization of Lemma 42.38.2 we conclude.
$\square$
Lemma 42.38.5. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $Y$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $\alpha $ be a $k$-cycle on $Y$. Then
\[ f^*(c_ j(\mathcal{E}) \cap \alpha ) = c_ j(f^*\mathcal{E}) \cap f^*\alpha \]
Proof.
Write $\alpha _ j = c_ j(\mathcal{E}) \cap \alpha $, so $\alpha _0 = \alpha $. By Lemma 42.38.2 we have
\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_ P(1))^ i \cap \pi ^*(\alpha _{r - i}) = 0 \]
in the chow group of the projective bundle $(\pi : P \to Y, \mathcal{O}_ P(1))$ associated to $\mathcal{E}$. Consider the fibre product diagram
\[ \xymatrix{ P_ X = \mathbf{P}(f^*\mathcal{E}) \ar[r]_-{f'} \ar[d]_{\pi _ X} & P \ar[d]^\pi \\ X \ar[r]^ f & Y } \]
Note that $\mathcal{O}_{P_ X}(1)$ is the pullback of $\mathcal{O}_ P(1)$. Apply flat pullback $(f')^*$ (Lemma 42.20.2) to the displayed equation above. By Lemmas 42.26.2 and 42.14.3 we see that
\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_{P_ X}(1))^ i \cap \pi _ X^*(f^*\alpha _{r - i}) = 0 \]
holds in the chow group of $P_ X$. By the characterization of Lemma 42.38.2 we conclude.
$\square$
Lemma 42.38.6. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 42.29.1. Then $c_ j(\mathcal{E}|_ D) \cap i^*\alpha = i^*(c_ j(\mathcal{E}) \cap \alpha )$ for all $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$.
Proof.
Write $\alpha _ j = c_ j(\mathcal{E}) \cap \alpha $, so $\alpha _0 = \alpha $. By Lemma 42.38.2 we have
\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_ P(1))^ i \cap \pi ^*(\alpha _{r - i}) = 0 \]
in the chow group of the projective bundle $(\pi : P \to X, \mathcal{O}_ P(1))$ associated to $\mathcal{E}$. Consider the fibre product diagram
\[ \xymatrix{ P_ D = \mathbf{P}(\mathcal{E}|_ D) \ar[r]_-{i'} \ar[d]_{\pi _ D} & P \ar[d]^\pi \\ D \ar[r]^ i & X } \]
Note that $\mathcal{O}_{P_ D}(1)$ is the pullback of $\mathcal{O}_ P(1)$. Apply the gysin map $(i')^*$ (Lemma 42.30.2) to the displayed equation above. Applying Lemmas 42.30.4 and 42.29.9 we obtain
\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_{P_ D}(1))^ i \cap \pi _ D^*(i^*\alpha _{r - i}) = 0 \]
in the chow group of $P_ D$. By the characterization of Lemma 42.38.2 we conclude.
$\square$
At this point we have enough material to be able to prove that capping with Chern classes defines a bivariant class.
Lemma 42.38.7. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_ X$-module of rank $r$. Let $0 \leq p \leq r$. Then the rule that to $f : X' \to X$ assigns $c_ p(f^*\mathcal{E}) \cap - : \mathop{\mathrm{CH}}\nolimits _ k(X') \to \mathop{\mathrm{CH}}\nolimits _{k - p}(X')$ is a bivariant class of degree $p$.
Proof.
Immediate from Lemmas 42.38.3, 42.38.4, 42.38.5, and 42.38.6 and Definition 42.33.1.
$\square$
This lemma allows us to define the Chern classes of a finite locally free module as follows.
Definition 42.38.8. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_ X$-module of rank $r$. For $i = 0, \ldots , r$ the $i$th Chern class of $\mathcal{E}$ is the bivariant class $c_ i(\mathcal{E}) \in A^ i(X)$ of degree $i$ constructed in Lemma 42.38.7. The total Chern class of $\mathcal{E}$ is the formal sum
\[ c(\mathcal{E}) = c_0(\mathcal{E}) + c_1(\mathcal{E}) + \ldots + c_ r(\mathcal{E}) \]
which is viewed as a nonhomogeneous bivariant class on $X$.
By the remark following Definition 42.38.1 if $\mathcal{E}$ is invertible, then this definition agrees with Definition 42.34.4. Next we see that Chern classes are in the center of the bivariant Chow cohomology ring $A^*(X)$.
Lemma 42.38.9. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_ X$-module of rank $r$. Then
$c_ j(\mathcal{E}) \in A^ j(X)$ is in the center of $A^*(X)$ and
if $f : X' \to X$ is locally of finite type and $c \in A^*(X' \to X)$, then $c \circ c_ j(\mathcal{E}) = c_ j(f^*\mathcal{E}) \circ c$.
In particular, if $\mathcal{F}$ is a second locally free $\mathcal{O}_ X$-module on $X$ of rank $s$, then
\[ c_ i(\mathcal{E}) \cap c_ j(\mathcal{F}) \cap \alpha = c_ j(\mathcal{F}) \cap c_ i(\mathcal{E}) \cap \alpha \]
as elements of $\mathop{\mathrm{CH}}\nolimits _{k - i - j}(X)$ for all $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$.
Proof.
It is immediate that (2) implies (1). Let $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$. Write $\alpha _ j = c_ j(\mathcal{E}) \cap \alpha $, so $\alpha _0 = \alpha $. By Lemma 42.38.2 we have
\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_ P(1))^ i \cap \pi ^*(\alpha _{r - i}) = 0 \]
in the chow group of the projective bundle $(\pi : P \to Y, \mathcal{O}_ P(1))$ associated to $\mathcal{E}$. Denote $\pi ' : P' \to X'$ the base change of $\pi $ by $f$. Using Lemma 42.34.5 and the properties of bivariant classes we obtain
\begin{align*} 0 & = c \cap \left(\sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_ P(1))^ i \cap \pi ^*(\alpha _{r - i})\right) \\ & = \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_{P'}(1))^ i \cap (\pi ')^*(c \cap \alpha _{r - i}) \end{align*}
in the Chow group of $P'$ (calculation omitted). Hence we see that $c \cap \alpha _ j$ is equal to $c_ j(f^*\mathcal{E}) \cap (c \cap \alpha )$ by the characterization of Lemma 42.38.2. This proves the lemma.
$\square$
Comments (2)
Comment #5777 by Zongzhu Lin on
Comment #5785 by Johan on