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 ith 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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