Let $X$ be a smooth projective variety. With rational coefficients, the Grothendieck-Riemann-Roch theorem implies that the Chern character induces a canonical isomorphism $\mathrm{ch} : \mathrm{K}_0(X) \otimes \mathbf{Q} \stackrel{\sim}{\longrightarrow} \mathrm{CH}^*(X) \otimes \mathbf{Q}$. What are counterexamples of this statement for integral coefficients?
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did you ever find a counterexample? – 54321user May 23 '17 at 02:06
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Coming across this after a long time (4 years I guess). It doesn't really make sense with integral coefficients (the chern character is still a map from $K_0(X)$ to $CH^*(X)\otimes \mathbf{Q}$ -- there are a lot of instances when this map is not an isomorphism, i.e. affine or projective space). The statement usually one asks is when is the ``inverse" an isomorphism. In quotes because it's actually a map from $CH(X)\rightarrow \text{gr}_\tau G_0(X)$ sending $[V]\rightarrow [\mathcal{O}_V]$. This map is always a surjection and the GRR implies the kernel is torsion, c.f. example 15.3.6 in Fulton – Eoin Oct 13 '17 at 04:20