The cohomology of the finite general linear group
Let F be a finite field of characteristic p, \(\ell\) a prime \(\neq p\), and G the F-points of the general linear group of n by n matrices. The author extends a result of Quillen's by computing the cohomology of G with coefficients in \({\mathbb{Z}}/\ell\) k\({\mathbb{Z}}\), when card(F) is congruent to 1 mod \(\ell\) k (resp. \(\ell^{k+1})\) if \(\ell\) is odd (resp. 2). The cohomology is the product of a free polynomial and a free exterior algebra over \({\mathbb{Z}}/\ell\) k, with generators in degrees 2,4,...,2n and 1,3,...,2n-1 resp. From here it is not too hard to deduce Quillen's original theorem for \({\mathbb{Z}}/\ell\) coefficients (with no restriction except \(\ell \neq p)\) and results where F is replaced by its algebraic closure. The methods involve Quillen's detection theorem to get upper bounds on the cohomology and transfer-restriction arguments from parabolic subgroups of G to construct the cohomology classes that are claimed to exist.
scientific article; zbMATH DE number 4069189
scientific article