Let G be the group scheme ${\mathrm{SL}}_{d+1}$ 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.

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关于某些标志方案上的线束的同调

设G为分组方案${\mathrm{SL}}_{d+\mathrm{1个}}$ 超过 $\mathbb{ž}$令Q为对应于简单根的抛物子群方案${\alpha }_{\mathrm{2个}}，\cdots ，{\alpha }_{d-\mathrm{1个}}$。然后$G/问$ 是个 $\mathbb{ž}$-部分标志的方案 $\{d_{\mathrm{1个}}\subset H_d\subset \mathrm{伏特}\}$。我们将在该标志方案上计算线束的同调模块。我们将证明，只有非平凡的那些是具有多项式系数的某些矩阵的核或核的同构。