We give a description of the cohomology groups of the structure sheaf on smooth compactifications \(\overline{X}(w)\) of Deligne–Lusztig varieties X(w) for \(\textrm{GL}_n\) , for all elements w in the Weyl group. As a consequence, we obtain the \(\textrm{mod}\ p^m\) and integral p-adic étale cohomology of \(\overline{X}(w)\) . Moreover, using our result for \(\overline{X}(w)\) and a spectral sequence associated to a stratification of \(\overline{X}(w)\) , we deduce the \(\textrm{mod}\ p^m\) and integral p-adic étale cohomology with compact support of X(w). In our proof of the main theorem, in addition to considering the Demazure–Hansen smooth compactifications of X(w), we show that a similar class of constructions provide smooth compactifications of X(w) in the case of \(\textrm{GL}_n\) . Furthermore, we show in the appendix that the Zariski closure of X(w), for any connected reductive group G and any w, has pseudo-rational singularities.
