As far as I understand, for a smooth variety $X$ its motivic cohomology could be described as the corresponding piece of the $\gamma$-filtration of (Quillen's) $K^*(X)$; this is completely true for $\mathbb{Q}$-coefficients, and true up to bounded denominators for $\mathbb{Z}$-coefficients.

My question is: is there a similar result for singular varieties? Here for motivic cohomology I would like to take $Hom_{DM}(M(X),\mathbb{Z}(p)[q]$; $DM$ is the category of Voevodsky's motives, and $M(X)$ is the motif of $X$ (I don't want to take the motif with compact support instead). These cohomology theories satisfy cdh-descent, but have no easy descriptions in terms of complexes of algebraic cycles. Unfortunately, I don't know much about the $\gamma$-filtration of $K$-theory.

Actually, I would like to prove the following fact: if a morphism $X\to Y$ of varieties induces an isomorphism for $K^*$, then the exponents of the kernels and cokernels of the corresponding morphisms for motivic cohomology are bounded (by a constant that depends only only on the dimensions of $X$ and $Y$). Any hints and/or references would be very welcome!


1 Answer 1


The precise relationship between K-theory and motivic cohomology for smooth schemes is the (analog of the) Atiya-Hirzebruch spectral sequence. This generalizes to non-smooth schemes if one uses K'-theory and higher Chow groups.

It is easy to see that motivic cohomology cannot be used to recover K-theory, because it does not detect nilpotents. For example, $k[t]/t^2$ has K-theory different from $k$, but motivic cohomology is the same. cdh-descent than gives examples for reduced schemes (look at a cusp).

There are some approaches (Bloch-Esnault) to beef up cycles to get results in the above examples.

As for your original question, recover motivic cohomology from K-theory, this might be possible, but I have no result to offer.


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