Abstract :
Let R be a field and Ω an n-element set. For k ⩽ n consider the R-vector space Mk with k-element subsets of Ω as basis. The inclusion map ∂ : Mk → Mk − 1 is the linear map defined on this basis through ∂(Δ) := Γ1 + Γ2 + ⋯ + Γk, where the Γi are the (k − 1)-element subsets of Δ. Thus, we obtain a chain 0←M0←←M1←…←Mk−1←Mk←Mk+1←…←Mn0 of inclusion maps. In non-zero characteristic such chains have interesting homological properties which have been examined in earlier papers (Mnukhin and Siemons, 1996). The purpose of this note is to study generators for the homology modules when R is a field of characteristic p ⩾ n.
