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](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.154.9304&rep=rep1&type=pdf). 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](https://conf.math.illinois.edu/K-theory/0206/adams.pdf) 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.



Grayson filtration and weight filtration
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). 


