On the Friedlander-Milnor conjecture for groups of small rank (Q2902058)

The main theorem in this paper is the following weak version of the Friedlander-Milnor conjecture:NEWLINENEWLINE Theorem. Let \(F\) be a separably closed field and \(G\) be any reductive algebraic \(F\)-group and let \(\ell\) be a prime different from the characteristic char\((F)\). Then the natural homomorphism NEWLINE\[NEWLINE H^*_{\text{ét}}(BG; \mathbb {Z}/{\ell})\to H^*(B(G(\Delta_F^\bullet ));\mathbb{Z}/{\ell}) NEWLINE\]NEWLINE is an isomorphism. In the above, on the left hand side \(BG\) denotes the classifying space of \(G\), and \(H^*_{\text{ét}}\) is étale cohomology with coefficients in \(\mathbb{Z}/{\ell}\). On the right hand side, \(\Delta_F^\bullet\) denotes the cosimplicial affine \(F\)-scheme defined by \(\Delta^n_F:=\)Spec\((F[T_0,\ldots ,T_n ]/\sum_iT_{i-1})\) with the standard cosimplicial structure. Moreover, \(G(\Delta^\bullet_F)\) is the simplicial group obtained by evaluating \(G\) on \(\Delta^\bullet_F\), and \(BG(\Delta^\bullet_F)\) denotes its classifying space.NEWLINENEWLINEAn important concept that implies the Friedlander conjecture is: a group \(G\) satisfies the homotopy invariance of group cohomology at \(\ell\) if for any integer \(n\geq1\) the natural ring homomorphism NEWLINE\[NEWLINE H^*(BG(F[T_1,\ldots ,T_n];\mathbb{Z}/{\ell})\to H^*(BG(F);\mathbb{Z}/{\ell}) NEWLINE\]NEWLINE is an isomorphism.NEWLINENEWLINEProposition. Let \(F\) be a separably closed field, \(G\) be a reductive algebraic \(F\)-group and \(\ell\) be a prime different from char\((F)\). Assume \(G\) satisfies the homotopy invariance of group cohomology at \(\ell\), then the map of simplicial sets NEWLINE\[NEWLINE BG(F)\to B(G(\Delta^\bullet_F)) NEWLINE\]NEWLINE induces an isomorphism on mod \(\ell\)-homology and consequently the Friedlander conjecture holds for \(G\).NEWLINENEWLINEExamples of groups that satisfy the above assumptions (also proved in the paper) are: let \(F\) be as above and \(G\) be a split simple algebraic group of type \(\mathrm{SL}_3, \mathrm{SO}_5, \mathrm{G}_2\) or \(\mathrm{SL}_4\).NEWLINENEWLINEFor the entire collection see [Zbl 1245.00031].