Given a (thick) irreducible spherical building \(\Omega \), we establish a bound on the difference between the generating rank and the embedding rank of its long root geometry and the dimension of the corresponding Weyl module, by showing that this difference does not grow when taking certain residues of \(\Omega \) (in particular the residue of a vertex corresponding to a point of the long root geometry, but also other types of vertices occur). We apply this to the finite case to obtain new results on the generating rank of mainly the exceptional long root geometries, answering an open question by Cooperstein about the generating ranks of the exceptional long root subgroup geometries. We completely settle the finite case for long root geometries of type \({{\textsf{A}}}_n\), and the case of type \(\mathsf {F_{4,4}}\) over any field with characteristic distinct from 2 (which is not a long root subgroup geometry, but a hexagonic geometry).

给定一个（厚的）不可约球形建筑\(\Omega \)，我们在其长根几何结构的生成等级和嵌入等级与相应 Weyl 模块的维度之间的差异建立一个界限，通过证明这种差异确实当取\(\Omega \)的某些残数时不会增长（特别是对应于长根几何体的一个点的顶点的残数，但也出现其他类型的顶点）。我们将其应用于有限情况，以获得主要是异常长根几何形状的生成等级的新结果，回答了库珀斯坦关于异常长根子群几何形状的生成等级的开放问题。我们完全解决了\({{\textsf{A}}}_n\)类型的长根几何形状的有限情况，以及任何字段上\(\mathsf {F_{4,4}}\)类型的情况与 2 不同的特征（不是长根子群几何，而是六边形几何）。