On modular homology in projective space

For a vector space V over GF(q) let Lk be the collection of subspaces of dimension k. When R is a field let Mk be the vector space over it with basis Lk. The inclusion map∂:Mk→Mk−1 then is the linear map defined on this basis via ∂(X) ∑Y where the sum runs over all subspaces of co-dimension 1 in X. This gives rise to a sequence which has interesting homological properties if R has characteristic p>0 not dividing q. Following on from earlier papers we introduce the notion of π-homological, π-exact and almostπ-exact sequences where π=π(p,q) is some elementary function of the two characteristics. We show that and many other sequences derived from it are almost π-exact. From this one also obtains an explicit formula for the Brauer character on the homology modules derived from . For infinite-dimensional spaces we give a general construction which yields π-exact sequences for finitary ideals in the group ring RPΓL(V).