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There is a well-known Quillen's localization sequence for (algebraic) K-theory: $\dots\to K_p^Y(X)\to K_p(X)\to K_p(X-Y)\to \dots$, where $Y\to X$ is a closed embedding of schemes. Now suppose that $ …

It is well-known (and was proved by Gabber?): if $R$ is a regular henselian local ring containing a field of characteristic prime to $l$, $k$ is its residue field, then $K_\ast(R,\mathbb{Z}/l)\cong K …

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 Gill …

What is the $K$-theory of the category of coherent sheaves on the infinite (countable) dimensional projective space over a field? As far as I know, $K$-theory is oriented; hence this theory should be …

The Chern class yields an isomorphism $K_0(X)\otimes \mathbb Q\cong \bigoplus_{i\ge 0} Chow^i(X)\otimes \mathbb Q$ (for a smooth variety $X$ over a field?), whereas the latter group is isomorphic to $ …

The "motivic motivation" is that by idempotent completing correspondences over a finite field one obtains a category of homological motives where Kunneth decompositions of diagonals are available. Mo …

There are two ways to define Borel-Moore motivic homology (of schemes) with rational coefficients: one should either consider certain complexes of algebraic cycles, or the $\gamma$-filtrations of Quil …

I'm not sure that it is possible to compress the big picture into one answer; yet I will try to give a hint. Firstly, one can hardly hope to have a "reasonable" motivic $t$-structure for motives with …