Mod $\ell$ cohomology of some Deligne-Lusztig varieties for $\operatorname{GL}_n(q)$
In this article, we study the mod $\ell$ cohomology of some Deligne-Lusztig varieties for $\operatorname{GL}_n(q)$. We prove that the cohomology groups of these varieties are torsion-free under some conditions on the characteristic. Under the torsion-free assumption we can compute the cohomology groups explicitly and we prove that the cohomology complex satisfies partial-tilting condition, which is one of the necessary conditions in the geometric version of Broué's abelian defect group conjecture.
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