The Stacks project

Remark 42.23.5. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. We will see later (in Lemma 42.69.3) that the map

\[ \mathop{\mathrm{CH}}\nolimits _ k(X) \longrightarrow K_0(\textit{Coh}_{k + 1}(X)/\textit{Coh}_{\leq k - 1}(X)) \]

of Lemma 42.23.4 is injective. Composing with the canonical map

\[ K_0(\textit{Coh}_{k + 1}(X)/\textit{Coh}_{\leq k - 1}(X)) \longrightarrow K_0(\textit{Coh}(X)/\textit{Coh}_{\leq k - 1}(X)) \]

we obtain a canonical map

\[ \mathop{\mathrm{CH}}\nolimits _ k(X) \longrightarrow K_0(\textit{Coh}(X)/\textit{Coh}_{\leq k - 1}(X)). \]

We have not been able to find a statement or conjecture in the literature as to whether this map is should be injective or not. It seems reasonable to expect the kernel of this map to be torsion. We will return to this question (insert future reference).


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