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For a simplicial complex 2 and coefficient domain F let F2 be the F-module with basis 2. We investigate the inclusion map given by : { [ _ 1 +_ 2 +_ 3 + } } } +_ k which assigns to every face { the sum of the co-dimension 1 faces contained in {. When the coefficient domain is a field of characteristic p>0 this map gives rise to homological sequences. We investigate this modular homology for shellable complexes. Academic Press

Let 0 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 0 as basis. The set inclusion map : M k Ä M k&1 is the homomorphism which associates to a k-element subset 2 the sum (2)=1 1 +1 2 + } } } +1 k of all its (k&1)-element subsets 1 i . In this paper we study the chainarising from . We introduce the notion of p-exactness for a sequence and show that any interval of (*) not including M nÂ2 or M n+1Â2 respectively, is p-exact for any prime p>0. This result can be extended to various submodules and quotient modules, and we give general constructions for permutation groups on 0 of order not divisible by p. If an interval of (*) , or an equivalent sequence arising from a permutation group on 0, does include the middle term then proper homologies can occur. In these cases we have determined all corresponding Betti numbers. A further application are p-rank formulae for orbit inclusion matrices. AcademicPress, Inc.

Among shellable complexes a certain class is shown to have maximal modular homology, and these are the so-called saturated complexes. We show that certain conditions on the links of the complex imply saturation. We prove that Coxeter complexes and buildings are saturated. © 2002 Elsevier Science (USA)

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