The cohomology of the nilCoxeter algebra

The nilCoxeter algebra NSₙ of the symmetric group Sₙ is the algebra over Z with generators Yᵢ (1 i n-1), satisfying the braid relations YᵢYᵢ₊₁Yᵢ=Yᵢ₊₁YᵢYᵢ₊₁, YᵢYⱼ=YⱼYᵢ (|j-i| 2), together with the relations Yᵢ²=0. We describe an explicit presentation for the cohomology ring Z=Ext^*NSₙ(Z,Z), with $n-i$ new generators in degree i for $0< i<n$, and all relations are quadratic. We show that this Ext ring is Z-free, and that it is a semiprime Noetherian affine polynomial identity (PI) ring with Poincar\'e series 1/(1-t)ⁿ⁻¹ and PI degree 2ⁿ⁻². For any field of coefficients k, we show that Ext^*kNSₙ(k,k) is k⊗Z Z. Similar results hold for other finite Coxeter types.