THE MODULAR HOMOLOGY OF INCLUSION MAPS AND GROUP-ACTIONS

Let Omega be a finite set of n elements, R a ring of characteristic p > 0 and denote by M(k) the R-module with k-element subsets of Omega as basis. The set inclusion map partial derivative:M(k) --> M(k-1) is th e homomorphism which associates to a k-element subset Delta the sum pa rtial derivative(Delta) = Gamma(1) + Gamma(2) + ... + Gamma(k) of all its (k - 1)-element subsets Gamma(i). In this paper we study the chain 0 <-- M(0) <-- M(1) <-- M(2) ... M(k) <-- M(k+1) <-- M(k+2)... () ar ising from partial derivative. We introduce the notion of p-exactness for a sequence and show that any interval of () not including M(n/2) or M(n+1/2) respectively, is p-exact for any prime p > 0. This result can be extended to various submodules and quotient modules, and we giv e general constructions for permutation groups on Omega of order not d ivisible by p. If an interval of (), or an equivalent sequence arisin g from a permutation group on Omega, does include the middle term then proper homologies can occur. In these cases we have determined all co rresponding Betti numbers. A further application are p-rank formulae f or orbit inclusion matrices. (C) 1996 Academic Press, Inc.