82.28 The Chern classes of a vector bundle
This section is the analogue of Chow Homology, Sections 42.37 and 42.38. However, contrary to what is done there, we directly define the Chern classes of a vector bundle as bivariant classes. This saves a considerable amount of work.
Lemma 82.28.1. In Situation 82.2.1 let $X/B$ be good. 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 space bundle associated to $\mathcal{E}$. For every morphism $X' \to X$ of good algebraic spaces over $B$ there are unique maps
\[ c_ i(\mathcal{E}) \cap - : \mathop{\mathrm{CH}}\nolimits _ k(X') \longrightarrow \mathop{\mathrm{CH}}\nolimits _{k - i}(X'),\quad i = 0, \ldots , r \]
such that for $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X')$ we have $c_0(\mathcal{E}) \cap \alpha = \alpha $ and
\[ \sum \nolimits _{i = 0, \ldots , r} (-1)^ i c_1(\mathcal{O}_{P'}(1))^ i \cap (\pi ')^*\left(c_{r - i}(\mathcal{E}) \cap \alpha \right) = 0 \]
where $\pi ' : P' \to X'$ is the base change of $\pi $. Moreover, these maps define a bivariant class $c_ i(\mathcal{E})$ of degree $i$ on $X$.
Proof.
Uniqueness and existence of the maps $c_ i(\mathcal{E}) \cap -$ follows immediately from Lemma 82.27.2 and the given description of $c_0(\mathcal{E})$. For every $i \in \mathbf{Z}$ the rule which to every morphism $X' \to X$ of good algebraic spaces over $B$ assigns the map
\[ t_ i(\mathcal{E}) \cap - : \mathop{\mathrm{CH}}\nolimits _ k(X') \longrightarrow \mathop{\mathrm{CH}}\nolimits _{k - i}(X'),\quad \alpha \longmapsto \pi '_*(c_1(\mathcal{O}_{P'}(1))^{r - 1 + i} \cap (\pi ')^*\alpha ) \]
is a bivariant class1 by Lemmas 82.26.4, 82.26.5, and 82.26.7. By Lemma 82.27.1 we have $t_ i(\mathcal{E}) = 0$ for $i < 0$ and $t_0(\mathcal{E}) = 1$. Applying pushforward to the equation in the statement of the lemma we find from Lemma 82.27.1 that
\[ (-1)^ r t_1(\mathcal{E}) + (-1)^{r - 1}c_1(\mathcal{E}) = 0 \]
In particular we find that $c_1(\mathcal{E})$ is a bivariant class. If we multiply the equation in the statement of the lemma by $c_1(\mathcal{O}_{P'}(1))$ and push the result forward to $X'$ we find
\[ (-1)^ r t_2(\mathcal{E}) + (-1)^{r - 1} t_1(\mathcal{E}) \cap c_1(\mathcal{E}) + (-1)^{r - 2} c_2(\mathcal{E}) = 0 \]
As before we conclude that $c_2(\mathcal{E})$ is a bivariant class. And so on.
$\square$
Definition 82.28.2. In Situation 82.2.1 let $X/B$ be good. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. 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 82.28.1. 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 $X$.
For convenience we often set $c_ i(\mathcal{E}) = 0$ for $i > r$ and $i < 0$. By definition we have $c_0(\mathcal{E}) = 1 \in A^0(X)$. Here is a sanity check.
Lemma 82.28.3. In Situation 82.2.1 let $X/B$ be good. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. The first Chern class of $\mathcal{L}$ on $X$ of Definition 82.28.2 is equal to the bivariant class of Lemma 82.26.4.
Proof.
Namely, in this case $P = \mathbf{P}(\mathcal{L}) = X$ with $\mathcal{O}_ P(1) = \mathcal{L}$ by our normalization of the projective bundle, see Section 82.27. Hence the equation in Lemma 82.28.1 reads
\[ (-1)^0 c_1(\mathcal{L})^0 \cap c^{new}_1(\mathcal{L}) \cap \alpha + (-1)^1 c_1(\mathcal{L})^1 \cap c^{new}_0(\mathcal{L}) \cap \alpha = 0 \]
where $c_ i^{new}(\mathcal{L})$ is as in Definition 82.28.2. Since $c_0^{new}(\mathcal{L}) = 1$ and $c_1(\mathcal{L})^0 = 1$ we conclude.
$\square$
Next we see that Chern classes are in the center of the bivariant Chow cohomology ring $A^*(X)$.
Lemma 82.28.4. In Situation 82.2.1 let $X/B$ be good. Let $\mathcal{E}$ be a locally free $\mathcal{O}_ X$-module of rank $r$. Then $c_ j(\mathcal{L}) \in A^ j(X)$ commutes with every element $c \in A^ p(X)$. 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.
Let $X' \to X$ be a morphism of good algebraic spaces over $B$. Let $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X')$. Write $\alpha _ j = c_ j(\mathcal{E}) \cap \alpha $, so $\alpha _0 = \alpha $. By Lemma 82.28.1 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 $(X' \to X)^*\mathcal{E}$. Applying $c \cap -$ and using Lemma 82.26.8 and the properties of bivariant classes we obtain
\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_{P'}(1))^ i \cap \pi ^*(c \cap \alpha _{r - i}) = 0 \]
in the Chow group of $P'$. Hence we see that $c \cap \alpha _ j$ is equal to $c_ j(\mathcal{E}) \cap (c \cap \alpha )$ by the uniqueness in Lemma 82.28.1. This proves the lemma.
$\square$
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