ac.commutative algebra - Filtration over tensor product

ag.algebraic geometry – Comparing $Okay$-cohomology teams and weight filtration on the $Okay$-groups retort

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ag.algebraic geometry – Comparing $Okay$-cohomology teams and weight filtration on the $Okay$-groups

The second web page of the Quillen-Brown-Gersten is within the following design:
$$E_2^{p,q}=H^{p}(X, mathcal{Okay}_{-q})Rightarrow K_{-q-p}(X)$$

Here $mathcal{Okay}_n$ is sheafification of the $Umapsto K_n(U)$ with respect to the Zariski topology. Is this spectral sequence anticipated most often to degenerate on the second web page? How distant is $H^{p-q}(X, mathcal{Okay}_p)$ from $K_q(X)^{(p)}$? (Maybe rationally.)
A unostentatious occasion would breathe evaluating $K_1(X)^{(2)}$ and $H^{1}(X, mathcal{Okay}_2)$. Unfortunately I do not know mighty examples to check them. Are they conjectural to breathe the similar rationally? Are there examples so you possibly can compute each of them?

In this bespeak web page 119 a inescapable group $Q(G)$ is launched and it’s proved to breathe isomorphic to $H^{1}(X, mathcal{Okay}_2)$ for $X$ a merely linked algebraic group. Unfortunately I assassinate not know many strategies to as an example compute $K_1(X)^{(2)}$, I used to be solely in a position to assassinate it for $SL_2$ which each coincide with $mathbb{Z}$ or $mathbb{Q}$ rationally. If the retort to my first query is just not limpid, are there strategies to compute $K_1(X)^{(2)}$ for $X$ simply-connected algebraic group?

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