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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 $X$ is regular (and excellent of finite dimension, if needed). Another well-known fact is that (in this case) the relative K-theory group $K_p^Y(X)$ is isomorphic to $K'_p(Y)$ (some authors denote this by $G_p(Y)$; note that $Y$ is not necessarily regular!).

Now, I tensor this long exact sequence by $\mathbb{Q}$. Can I consider the $i$-th graded piece of the $\gamma$-filtration for this long exact sequence? Certainly, $K^{}(X)\otimes \mathbb{Q}$ and $K^*(X-Y)\otimes \mathbb{Q}$ are endowed with $\gamma$-filtration, but I am not quite sure about $K'_p(Y)\otimes \mathbb{Q}$ (one of my problems here is that I am interested in quite a general situation). Also, could I say that the $i$-th level of the $\gamma$-filtration for $K'_p(Y)$ is some (which one??) level of its niveau filtration?

Which references are most appropriate for these matters? I believe that for rational coefficients these things are easier than for integral ones.

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up vote 6 down vote accepted

The basic reference is Soule's paper "Operations en K-theorie algebrique" (Can. J. Math. 37 (1985) 488-550). Essentially, there is a grading on $K'$-theory (with rational coefficients) enabling one to interpret the localisation sequence as the long exact sequence of motivic homology. The filtration comes from the $\gamma$-filtration on $K_*^Y(X)$, the $K$-theory with support.

Tamme's article in the Beilinson conjectures book ( is a good survey of this, but he sticks to the case of schemes over a field.

I am not sure I understand your final question, since even for $K$-theory of nice schemes (e.g. spectra of fields) the $\gamma$- and niveau filtrations don't agree except for $K_0$.

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Even when tensored by $\mathbb{Q}$? Thank you! – Mikhail Bondarko Jul 24 '10 at 15:39
Also, could you give a reference for the $K_0$-case? – Mikhail Bondarko Jul 24 '10 at 15:42
SGA6 is the bible for this (the introductory Expose 0 by Grothendieck summarises what's inside nicely) – Tony Scholl Jul 24 '10 at 16:23
Another good reference for the gamma and dimension filtration is Gillet's article for the K-theory handbook, available at It contains two different proofs for smooth schemes over a field. – Timo Schürg Jul 24 '10 at 22:50
Do I understand correctly that the $i$-th level of the filtration for $K^*(X)$ (and so also for $K^Y(X)$) by definition becomes the $d-i$-th level of the filtration for $K'(Y)$, where $d$ is the dimension of $X$? Is this convention (for the levels of the filtrations) canonical? – Mikhail Bondarko Jul 25 '10 at 11:33

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