The Modular Homology of Inclusion Maps and Group Actions

LetΩbe a finite set ofnelements,Ra ring of characteristicp>0 and denote byMktheR-module withk-element subsets of 0 as basis. The set inclusion map ∂: Mk→Mk−1is the homomorphism which associates to ak-element subset 2 the sum ∂(Δ)=Γ1+Γ2+…+Γkof all its (k−1)-element subsetsΓi. In this paper we study the chain[formula]arising from ∂. We introduce the notion ofp-exactness for a sequence and show that any interval of (*) not includingMn/2orMn+1/2respectively, isp-exact for any primep>0. This result can be extended to various submodules and quotient modules, and we give general constructions for permutation groups onΩof order not divisible byp. If an interval of (*) , or an equivalent sequence arising from a permutation group onΩ, does include the middle term then proper homologies can occur. In these cases we have determined all corresponding Betti numbers. A further application arep-rank formulae for orbit inclusion matrices.