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Remark 42.45.1. In the discussion above we have defined the components of the Chern character $ch_ p(\mathcal{E}) \in A^ p(X) \otimes \mathbf{Q}$ of $\mathcal{E}$ even if the rank of $\mathcal{E}$ is not constant. See Remarks 42.38.10 and 42.43.5. Thus the full Chern character of $\mathcal{E}$ is an element of $\prod _{p \geq 0} (A^ p(X) \otimes \mathbf{Q})$. If $X$ is quasi-compact and $\dim (X) < \infty$ (usual dimension), then one can show using Lemma 42.34.6 and the splitting principle that $ch(\mathcal{E}) \in A^*(X) \otimes \mathbf{Q}$.