We compute the cohomology $H^*(mathcal{H},k)=mathrm{Ext}^*_{mathcal{H}}(k,k)$ where $mathcal{H}=mathcal{H}(n,q)$ is the Hecke algebra of the symmetric group $mathfrak{S}_n$ at a primitive $ell$th root of unity $q$, and $k$ is a field of characteristic zero. The answer is particularly interesting when $ell=2$, which is the only case where it is not graded commutative. We also carry out the corresponding computation for Hecke algebras of type $B_n$ and $D_n$ when $ell$ is odd.
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