The Stacks project

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

  1. $ch^{dR}$ is a ring map for all $X$,

  2. 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

  3. given $\mathcal{L} \in \mathop{\mathrm{Pic}}\nolimits (X)$ we have $ch^{dR}([\mathcal{L}]) = \exp (c_1^{dR}(\mathcal{L}))$.

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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