## 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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