Remark 42.56.5. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. The Chern character map defines a canonical ring map

$ch : K_0(\textit{Vect}(X)) \longrightarrow \prod \nolimits _{i \geq 0} A^ i(X) \otimes \mathbf{Q}$

by sending a generator $[\mathcal{E}]$ on the left hand side to $ch(\mathcal{E})$ and extending additively. This is well defined by Lemma 42.45.2 and a ring homomorphism by Lemma 42.45.3.

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