On the cohomology of line bundles over certain flag schemes

TL;DR

This paper computes the cohomology of line bundles over certain flag schemes associated with SL_{d+1}, revealing that non-trivial cohomology modules are kernels or cokernels of matrices with multinomial coefficients.

Contribution

It provides explicit calculations of cohomology modules for line bundles on partial flag schemes of SL_{d+1}, identifying their structure as kernels or cokernels of specific matrices.

Findings

Non-trivial cohomology modules are kernels or cokernels of matrices with multinomial coefficients.

Explicit descriptions of cohomology modules for line bundles on partial flag schemes.

Cohomology calculations over integer schemes for algebraic groups.

Abstract

Let $G$ be the group scheme $\operatorname{SL}_{d+1}$ over $\mathbb{Z}$ and let $Q$ be the parabolic subgroup scheme corresponding to the simple roots $\alpha_{2},\cdots,\alpha_{d-1}$. Then $G/Q$ is the $\mathbb{Z} $-scheme of partial flags $\{D_{1}\subset H_{d}\subset V\}$. We will calculate the cohomology modules of line bundles over this flag scheme. We will prove that the only non-trivial ones are isomorphic to the kernel or the cokernel of certain matrices with multinomial coefficients.