Integral Clifford Theory and the Computation of Denominator Ideals

Abstract

Let R be a commutative ring. To each finitely presented R-module M one can associate an ideal, Fit_R(M), called the (zeroth) Fitting ideal of M . This ideal is always contained within the R-annihilator of M . Now let R be an integrally closed complete Noetherian local ring and let Λ be a (not necessarily commutative) R-order. A. Nickel generalised the notion of the Fitting ideal, providing a definition of the Fitting invariant for finitely presented modules M over Λ. In this case, to obtain the relation between the Fitting invariant of M and the annihilator of M in the centre of Λ, one must multiply the Fitting invariant of M by a certain ideal, H(Λ), of the centre of Λ, called the denominator ideal of Λ. H. Johnston and A. Nickel have formulated several bounds for the denominator ideal and have computed the denominator ideal for certain group rings. In this thesis, we prove a local-global principle for denominator ideals. We build upon the work of H. Johnston and A. Nickel to give improved bounds for the denominator ideal of Λ assuming some structural knowledge of Λ. We also build upon the work of P. Schmid and K. Roggenkamp to determine structural information about certain group rings. Finally, we use this structural information to compute the denominator ideal of group rings R[G], where G is a p-group with commutator subgroup of order p.