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The second page of the Quillen-Brown-Gersten is in the following form: $$E_2^{p,q}=H^{p}(X, \mathcal{K}_{-q})\Rightarrow K_{-q-p}(X)$$

Here $\mathcal{K}_n$ is sheafification of the $U\mapsto K_n(U)$ with respect to the Zariski topology. Is this spectral sequence expected in most cases to degenerate at the second page? How far away is $H^{p-q}(X, \mathcal{K}_p)$ from $K_q(X)^{(p)}$? (Maybe rationally.) A simple example would be comparing $K_1(X)^{(2)}$ and $H^{1}(X, \mathcal{K}_2)$. Unfortunately I don't know much examples to compare them. Are they supposed to be the same rationally? Are there examples so you can calculate both of them?

In this book page 119 a certain group $Q(G)$ is introduced and it is proved to be isomorphic to $H^{1}(X, \mathcal{K}_2)$ for $X$ a simply connected algebraic group. Unfortunately I do not know many methods to let's say calculate $K_1(X)^{(2)}$, I was only able to do it for $SL_2$ which both coincide with $\mathbb{Z}$ or $\mathbb{Q}$ rationally. If the answer to my first question is not clear, are there methods to calculate $K_1(X)^{(2)}$ for $X$ simply-connected algebraic group?

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    $\begingroup$ I'm not sure what you write for $K_1(X)^{(2)}$ here (I assume it's the associated quotient of the filtration on $K_1(X)$ but the index seems off a little bit, and sometimes people write $K_1(X)^{(i)}$ for something else entirely). Merkurjev has shown that the spectral sequence doesn't need to degenerate at the second page (mathoverflow.net/a/289803/65919). The sequence degenerates rationally at the $p+q=0$ line (this follows from a version of the Grothendieck-Riemann-Roch without denominators). $\endgroup$
    – Eoin
    Commented Feb 24, 2021 at 0:44
  • $\begingroup$ Thanks, $K_1(X)^{(i)}$ I meant weight $i$ part of the adams operation which makes sense rationally. Does the counter-example also work with rational coefficients? $\endgroup$
    – user127776
    Commented Feb 24, 2021 at 0:55
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    $\begingroup$ The counterexamlpe I linked does not provide a counterexample rationally. The weight-j part $K_i^{(j)}(X)$ of the adams operations on $K_i(X)_{\Bbb{Q}}$ is isomorphic with Bloch's Higher Chow groups with rational coefficients (numdam.org/item/AST_1994__226__235_0). The groups $H^i(X,\mathcal{K}_{i+j})$ are integrally isomorphic to Bloch's higher Chow groups (in some degrees) when $j=0,1,2$. I don't know if the isomorphism ``comes from the BGQ spectral sequence" in any natural way. $\endgroup$
    – Eoin
    Commented Feb 24, 2021 at 1:20

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