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I had a question that might be well-known but I'm not sure where to find it. Grayson defined a filtration on the algebraic $K$-theory of affine regular rings via commuting automorphisms which you can find here. You can define the same thing for general regular schemes. My question is that is it known that for a regular scheme (or smooth varieties over a field) the filtration coincides with the filtration coming from the Adams operation?

Walker here has proved it with the assumption of resolution of singularities. Suslin later on proved that the motivic spectral sequence corresponding to the Grayson filtration has the same $E^2$ page as the Friedlander-Suslin spectral sequence. But it obviously does not imply whether the Adams and Grayson filtrations coincide rationally or not.

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My impression is that Adams operations are "well known" to act coherently on all levels of the weight spectral sequence for K-theory (of smooth varieties); probably, this fact was established by Gillet and Soul´e. It easily follows that the filtration induced by this spectral sequence on rational K-theory is the Adams one. You may find an argument of this sort (that yields even more information) in the Yagunov's paper "Motivic cohomology spectral sequence and Steenrod operations", Compos. Math. 152 (2016), no. 10, 2113–2133.

Now I should add a few words on versions of weight (or motivic) spectral sequences. The Friedlander-Suslin version is not popular now, since it depends on certain doubtful arguments of Bloch and Lichtenbaum. However, there is the Grayson version, and the Voevodsky-Levine version, and they are known to be isomorphic (see the paper I cited for more detail).

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  • $\begingroup$ I had a slight confusion about these motivic filtrations. In order to get them there is a sheafification process. So let's say since we know weight zero part vanishes it implies that the map from weight>=1 part of grayson filtration to K-theory space, induces isomorphism on higher homotopy groups locally. Or does it also induce the isomorphism globally (at least rationally)? For example take a non-affine variety and consider the grayson filtration for it (Without any sheafification). $\endgroup$
    – user127776
    Commented Mar 29, 2020 at 18:46
  • $\begingroup$ In the Walker's paper that I have referred, he has proved that the naive grayson filtration without any sheafificaiton process (he works with $\mathbb{P}^1$ instead of $\mathbb{G}_m$) on any smooth variety over a field of char 0 is rationally the same as Adams filtration. $\endgroup$
    – user127776
    Commented Mar 29, 2020 at 18:51
  • $\begingroup$ I am not an expert, sorry. I think that rationally everything "should" commute with sheafifications, since the filtration splits and all morphisms between K-theory of various schemes "should" be compatible with these splittings. $\endgroup$
    – Mikhail Bondarko
    Commented Mar 30, 2020 at 17:00

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