# Homology modules in projective space

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Homology Modules in Projective Space Daniel John Smith A thesis submitted to the School of Mathematics of the University of East Anglia in partial fulfilment of the requirements for the degree of Doctor of Philosophy April 2007 Abstract For a Boolean algebra one may consider the set of subsets of size k to be the basis for a permutation module over a field which has prime characteristic p. By considering an inclusion map one can obtain sequences of maps on these modules. It turns out that these maps and sequences have certain homological properties. This theory goes back to the topologist W.Mayer in the 1940’s. One can extend this idea to projective space by taking the set of k-dimensional subspaces over a field of size q to be the basis for such permutation modules where q is a prime power not dividing p. It turns out that such sequences are also homological in a certain way. The homology modules in this case are modules over the finite general linear groups GL(n, q). We prove a decomposition formula for these homology modules in terms of GL(n − 1, q)-modules. Such a decomposition formula is known for the case of the Boolean algebra from S.Bell, P.Jones, and J.Siemons (J. Algebra 199, 1996, 556-580) and matches the result we prove here when we put q = 1. We prove some interesting results as direct consequences of this decomposition formula. These include showing that the homological sequence discussed above is almost exact, proving a rank formula for certain incidence matrices and giving a condition which ensures that the homology module is irreducible. We also look at representations associated to these homology modules and give an explicit description for an irreducible representation of PGL(2, q). 1 Contents 1 Introduction 5 1.1 Modular Homology in the Boolean Algebra . . . . . . . . . . . 5 1.2 Modular Homology in Projective Space . . . . . . . . . . . . . 7 1.3 Summary of Main Results . . . . . . . . . . . . . . . . .

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