50.21 Chern classes
The results proved so far suffice to use the discussion in Weil Cohomology Theories, Section 45.12 to produce Chern classes in de Rham cohomology.
Lemma 50.21.1. There is a unique rule which assigns to every quasi-compact and quasi-separated scheme $X$ a total Chern class
\[ c^{dR} : K_0(\textit{Vect}(X)) \longrightarrow \prod \nolimits _{i \geq 0} H^{2i}_{dR}(X/\mathbf{Z}) \]
with the following properties
we have $c^{dR}(\alpha + \beta ) = c^{dR}(\alpha ) c^{dR}(\beta )$ for $\alpha , \beta \in K_0(\textit{Vect}(X))$,
if $f : X \to X'$ is a morphism of quasi-compact and quasi-separated schemes, then $c^{dR}(f^*\alpha ) = f^*c^{dR}(\alpha )$,
given $\mathcal{L} \in \mathop{\mathrm{Pic}}\nolimits (X)$ we have $c^{dR}([\mathcal{L}]) = 1 + c_1^{dR}(\mathcal{L})$
The construction can easily be extended to all schemes, but to do so one needs to slightly upgrade the discussion in Weil Cohomology Theories, Section 45.12.
Proof.
We will apply Weil Cohomology Theories, Proposition 45.12.1 to get this.
Let $\mathcal{C}$ be the category of all quasi-compact and quasi-separated schemes. This certainly satisfies conditions (1), (2), and (3) (a), (b), and (c) of Weil Cohomology Theories, Section 45.12.
As our contravariant functor $A$ from $\mathcal{C}$ to the category of graded algebras will send $X$ to $A(X) = \bigoplus _{i \geq 0} H_{dR}^{2i}(X/\mathbf{Z})$ endowed with its cup product. Functoriality is discussed in Section 50.3 and the cup product in Section 50.4. For the additive maps $c_1^ A$ we take $c_1^{dR}$ constructed in Section 50.9.
In fact, we obtain commutative algebras by Lemma 50.4.1 which shows we have axiom (1) for $A$.
To check axiom (2) for $A$ it suffices to check that $H^*_{dR}(X \coprod Y/\mathbf{Z}) = H^*_{dR}(X/\mathbf{Z}) \times H^*_{dR}(Y/\mathbf{Z})$. This is a consequence of the fact that de Rham cohomology is constructed by taking the cohomology of a sheaf of differential graded algebras (in the Zariski topology).
Axiom (3) for $A$ is just the statement that taking first Chern classes of invertible modules is compatible with pullbacks. This follows from the more general Lemma 50.9.1.
Axiom (4) for $A$ is the projective space bundle formula which we proved in Proposition 50.14.1.
Axiom (5). Let $X$ be a quasi-compact and quasi-separated scheme and let $\mathcal{E} \to \mathcal{F}$ be a surjection of finite locally free $\mathcal{O}_ X$-modules of ranks $r + 1$ and $r$. Denote $i : P' = \mathbf{P}(\mathcal{F}) \to \mathbf{P}(\mathcal{E}) = P$ the corresponding incusion morphism. This is a morphism of smooth projective schemes over $X$ which exhibits $P'$ as an effective Cartier divisor on $P$. Thus by Lemma 50.15.7 the complex of log poles for $P' \subset P$ over $\mathbf{Z}$ is defined. Hence for $a \in A(P)$ with $i^*a = 0$ we have $a \cup c_1^ A(\mathcal{O}_ P(P')) = 0$ by Lemma 50.15.6. This finishes the proof.
$\square$
Working with schemes over $\mathbf{Q}$ we can construct a Chern character.
Lemma 50.21.3. There is a unique rule which assigns to every quasi-compact and quasi-separated scheme $X$ over $\mathbf{Q}$ a “chern character”
\[ ch^{dR} : K_0(\textit{Vect}(X)) \longrightarrow \prod \nolimits _{i \geq 0} H_{dR}^{2i}(X/\mathbf{Q}) \]
with the following properties
$ch^{dR}$ is a ring map for all $X$,
if $f : X' \to X$ is a morphism of quasi-compact and quasi-separated schemes over $\mathbf{Q}$, then $f^* \circ ch^{dR} = ch^{dR} \circ f^*$, and
given $\mathcal{L} \in \mathop{\mathrm{Pic}}\nolimits (X)$ we have $ch^{dR}([\mathcal{L}]) = \exp (c_1^{dR}(\mathcal{L}))$.
The construction can easily be extended to all schemes over $\mathbf{Q}$, but to do so one needs to slightly upgrade the discussion in Weil Cohomology Theories, Section 45.12.
Proof.
Exactly as in the proof of Lemma 50.21.1 one shows that the category of quasi-compact and quasi-separated schemes over $\mathbf{Q}$ together with the functor $A^*(X) = \bigoplus _{i \geq 0} H_{dR}^{2i}(X/\mathbf{Q})$ satisfy the axioms of Weil Cohomology Theories, Section 45.12. Moreover, in this case $A(X)$ is a $\mathbf{Q}$-algebra for all $X$. Hence the lemma follows from Weil Cohomology Theories, Proposition 45.12.4.
$\square$
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