Strouthos, I.; (2011) Stably free modules over group rings. Doctoral thesis , UCL (University College London).

Abstract

We study finitely generated stably free modules over group rings associated to particular groups, primarily using Milnor’s construction of projective modules over rings and Quillen’s Patching Theorem. For an odd prime number q, we study the stably free module category of the group ring of the dihedral group of order 2q over the ring of integer coefficient polynomials in one variable. We show that, over this group ring, every stably free module is free if every stably free module is free over a particular localisation of a cyclic algebra over the ring of algebraic integers corresponding to the prime q. Using this result, we show that every stably free module is free over the group ring of the dihedral group of order 6 over the ring of integer coefficient polynomials, and we later extend this result to all dihedral groups of order 2q. Furthermore, we consider a class of projective modules, which we regard as locally free, in a certain precise sense, and which are related to stably free modules under certain conditions. We show that, over the integral group ring of the direct product of a quaternion group of order 8 and the infinite cyclic group, there are infinitely many such locally free modules of rank 1. In addition, we show that, over the algebra of integer coefficient quaternions, every projective module is free. Finally this thesis includes a treatment of resolutions of indecomposable modules over the integral group ring of the dihedral group of order 6. By studying the irreducible integral representations of this dihedral group, we construct certain resolutions for which there is a notion of duality and diagonalisability in the constituent homomorphisms, via the syzygy operator.

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