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})
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
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