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Let $p\not =\ell$ be primes. We study the etale cohomology$H^{*}_{et}(BGL_n(\bF_{p^s});\bZ/{\ell})$ over the algebraically closed field$\bar \bF_p$ by using the stratification methods from Molina-Vistoli. Tocompute this cohomology, we use the Delinge-Lusztig variety.

The etale cohomology of the general linear group over a finite field and the Deligne and Lusztig variety