Bases of T-equivariant cohomology of Bott-Samelson varieties

Abstract

We construct combinatorial bases of the $T$-equivariant ($T$ is the maximal torus) cohomology $H_T^{\bullet }(\Sigma ,k)$ of the Bott-Samelson variety $\Sigma$ under some mild restrictions on the field of coefficients $k$. This bases allow us to prove the surjectivity of the restrictions $H_T^{\bullet }(\Sigma ,k)\to H_T^{\bullet }({\pi }^{-1}(x),k)$ and $H_T^{\bullet }(\Sigma ,k)\to H_T^{\bullet }(\Sigma \setminus {\pi }^{-1}(x),k)$, where $\pi :\Sigma \to G/B$ 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 $H_T^{\bullet }(\Sigma ,k)$ and restricting them to ${\pi }^{-1}(x)$ or $\Sigma \setminus {\pi }^{-1}(x)$ respectively. As an application, we calculate the cohomology of the costalk-to-stalk embedding for the direct image ${\pi }_{\ast }{\underline{k}}_{\Sigma }$. This algorithm avoids division by 2, which allows us to reestablish 2-torsion for parity sheaves in Braden's example.

References in corpus (1)

• The T-equivariant Cohomology of Bott-Samelson varieties

Cited by in corpus (1)

• Tensor product decompositions for cohomologies of Bott-Samelson varieties