Friedlander's generalized isomorphism conjecture

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Let GG be an algebraic group over an algebraically closed field kk, and let nn be a positive integer such that the characteristic of kk does not divide nn. The simplicial schemes BG(k)BG(k) and BGkBG_k are related by the natural map

BG(k)⟶BGk,BG(k)\longrightarrow BG_k,

where BG(k)BG(k) is regarded as discrete. Friedlander's generalized isomorphism conjecture. This map should induce an isomorphism

Heˊt⁡∙(BGk,Z/n)⟶H∙(BG(k),Z/n)=H∙(G(k),Z/n).H^\bullet_{\operatorname{\acute et}}(BG_k,\mathbb Z/n)\longrightarrow H^\bullet(BG(k),\mathbb Z/n)=H^\bullet(G(k),\mathbb Z/n).

This compares the cohomology of the algebraic simplicial scheme BGkBG_k with the group cohomology of the discrete group of kk-points. The paper presents it as Friedlander's conjecture; its resolution status is not specified in the source.

References

Primary source

Kevin P. Knudson and Mark E. Walker, “Homology of linear groups via cycles in BGX”, arXiv:math/0311362 (2003).

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