Universal presheaves on group geometries, and modular representations

AbstractWe extend the ideas of sheaf homology on buildings (M. A. Ronan and S. D. Smith, J. Algebra 96 (1985), 319–346) to more general geometries with transitive automorphism groups. The main technical result is the construction of a universal extension of certain partial sheaves. This guarantees a rich supply of sheaves on such geometries. Computation of zero-homology then affords representations of the group. We consider various applications of the technique, for example to modular irreducibles for sporadic simple groups