Integral Clifford Theory and the Computation of Denominator Ideals
Watson, David
Date: 4 May 2018
Publisher
University of Exeter
Degree Title
PhD in Mathematics
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. ...
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.
Doctoral Theses
Doctoral College
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