Lemma 45.14.1. Assume given (D0), (D1), and (D2') satisfying axioms (A1), (A2), (A3), and (A4). There is a unique rule which assigns to every smooth projective $X$ over $k$ a ring homomorphism

$ch^ H : K_0(\textit{Vect}(X)) \longrightarrow \prod \nolimits _{i \geq 0} H^{2i}(X)(i)$

compatible with pullbacks such that $ch^ H(\mathcal{L}) = \exp (c_1^ H(\mathcal{L}))$ for any invertible $\mathcal{O}_ X$-module $\mathcal{L}$.

Proof. Immediate from Proposition 45.12.4 applied to the category of smooth projective schemes over $k$, the functor $A : X \mapsto \bigoplus _{i \geq 0} H^{2i}(X)(i)$, and the map $c_1^ H$. $\square$

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