I have a somewhat vague question. We know that Adams operation gives a filtration on $K_i(X)\otimes \mathbb{Q}$ for the scheme $X$ such that the weight $j$ elements are isomorphic to higher Bloch Chow groups $CH^j(X,i)\otimes \mathbb{Q}$ for example if $i=0$ this will give us $CH^j(X)\otimes \mathbb{Q}$ which are co-dimension $i$ cycles modulo rational equivalence. For other adequate equivalence relations like numerical or algebraic equivalence of cycles is there something like the rational algebraic $K$-theory and some filtration similar to the Adams filtration that gives us the groups generated by cycles modulo those other equivalence relations?

Edit: For algebraic equivalence relation there is something called semi-topological $K$-theory which I guess has the same relation with the cycles mod algebraic equivalence. It is constructed only over complex numbers and constructing it over arbitrary fields is apparently an open problem. In lectures by Friedlander it is stated that there is a natural map from the algebraic $K$-theory to the semi-topological one an induces isomorphism mod $n$ (I guess $n$ is the dimension of the variety) if semi-topological $K$-theory is as I suppose it should be then is it true that it is expected this map to induce an isomorphism after tensoring by $\mathbb{Q}$ if we assume the algebraic and rational equivalences coincide on $X$?

  • $\begingroup$ I don't know how much has been studied in regards to higher Chow groups with respect to other equivalence relations but, for $i=0$ the answer is yes. But it's not great: the Chern character $ch:K(X)\otimes \mathbb{Q}\rightarrow \mathrm{CH}(X)\otimes \mathbb{Q}$ is an isomorphism (for some classes of $X$), and there's a surjection $\pi_{\sim}:\mathrm{CH}(X)\rightarrow \mathrm{CH}_{\sim}(X)$ for any other adequate equivalence relation $\sim$; so one can take $K(X)_{\mathbb{Q}}/\ker(\pi_{\sim}\circ ch)$ and then take the obvious filtration on that... $\endgroup$
    – Eoin
    Jul 8 '19 at 2:48

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