The étale cohomology of the general linear group over a finite field and the Dickson algebra (Q1753064)

Let \(p\) and \(l\) be two different primes and \(X\) be a smooth algebraic variety over a finite field \(k= \mathbb F_p\). Let \({H^*}_{\mathrm{et}} (X, \mathbb Z/l)\) be the étale cohomology of \(X\) over \(k\). It is known that the cohomology of the classifying space (Milnor space) \(BG\) of any algebraic group \(G\) can be computed by smooth quasiprojective algebraic varieties \(X_i\). By inverse limit over \(X_i\) the authors compute \[ {H^*}_{\mathrm{et}} (BGL_n (\mathbb F_q), \mathbb Z/ l) = \lim_i {H^*}_{et} (X_i, \mathbb Z/l) \] where \(q=p^s\). In this computation they also use the Dickson algebra.