On Modular Homology in the Boolean Algebra

Let Ω be a set,Ra ring of characteristicp>0, and denote byMktheR-module withk-element subsets of Ω as basis. Theset inclusion map∂: Mk→Mk−1is the homomorphism which associates to ak-element subset Δ the sum ∂(Δ)=Γ1+Γ2+···+Γkof all its (k−1)-element subsets Γi. In this paper we study the chain[formula]arising from ∂. We introduce the notion ofp-exactness for a sequence. If Ω is infinite we show that (∗) isp-exact for all prime characteristicsp>0. This result can be extended to various submodules and quotient modules, and we give general constructions arising from permutation groups with a finitary section. Two particular applications are the following: The orbit module sequence of such a permutation group on Ω isp-exact for every primep, and we give a formula for thep-rank of the orbit inclusion matrix if the group has finitely many orbits onk-element subsets.