nLab Friedlander-Milnor isomorphism conjecture

under construction

Statement

Let

• $k$ be an algebraically closed field;

• $G$ be a reductive algebraic group over $k$;

• $\ell$ a prime number invertible in $k$ ($\ell \neq char(k)$);

then the canonical morphism

$$H^\bullet_{et}(\mathbf{B}G, \mathbb{Z}/\ell\mathbb{Z}) \longrightarrow H^\bullet_{et}(\flat \mathbf{B}G, \mathbb{Z}/\ell\mathbb{Z}) = H^\bullet(B G(k), \mathbb{Z}/\ell\mathbb{Z})$$

(from the etale cohomology of the moduli stack of $G$ (the quotient stack $\mathbf{B}G \simeq \ast//G$) to the cohomology of its underlying discrete group of $k$-points) is an isomorphism.

References

Review includes

• Eric Friedlander, The Friedlander-Milnor conjecture (pdf)

• MO discussion

• Fabien Morel, On the homology of Lie groups made discrete, talk, 2012 (video)

Original articles include

• Eric Friedlander, Guido Mislin, Cohomology of classifying spaces of complex Lie groups and related discrete groups (pdf)