Bases of T-equivariant cohomology of Bott-Samelson varieties

2016/01/26 by Shchigolev, Vladimir
#Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1601.07146

Abstract

We construct combinatorial bases of the T-equivariant (T is the maximal torus) cohomology H^\bullet_T(Σ,k) of the Bott-Samelson variety Σ under some mild restrictions on the field of coefficients k. This bases allow us to prove the surjectivity of the restrictions H^\bullet_T(Σ,k)→ H^\bullet_T(π^-1(x),k) and H^\bullet_T(Σ,k)→ H^\bullet_T(Σ∖π^-1(x),k), where π:Σ→ G/B is the canonical resolution. In fact, we also construct bases of the targets of these restrictions by picking up certain subsets of certain bases of H^\bullet_T(Σ,k) and restricting them to π^-1(x) or Σ∖π^-1(x) respectively. As an application, we calculate the cohomology of the costalk-to-stalk embedding for the direct image π_*\underline k_Σ. This algorithm avoids division by 2, which allows us to reestablish 2-torsion for parity sheaves in Braden's example.

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