On Modular Homology of Simplicial Complexes: Shellability

For a simplicial complex Δ and coefficient domain F let FΔ be the F-module with basis Δ. We investigate the inclusion map given by ∂: τ↦σ1+σ2+σ3+…+σk which assigns to every face τ the sum of the co-dimension 1 faces contained in τ. When the coefficient domain is a field of characteristic p>0 this map gives rise to homological sequences. We investigate this modular homology for shellable complexes.