Partial Trace Ideals, The Conductor and Berger's Conjecture

The study of the vanishing of the torsion submodule of a finitely generated module M over a Noetherian ring R, has been a topic of high interest among algebraists. There exist quite a few open questions in specific setups, for example, the Huneke-Wiegand Conjecture, the Berger Conjecture, etc. This dissertation is aimed at providing a new approach to this torsion problem by introducing an invariant h(M) for any such module M over a one dimensional Noetherian local domain R. This invariant is based on a partial trace ideal of M. We show that the torsion of M is non-zero, whenever R is the quotient of a regular local ring with h(M) satisfying a suitable upper bound which is dependent on the embedding dimension of R and the properties of the defining ideal of R. We further make an independent study of h(M) for any finitely generated module M and establish links with the conductor ideal of R. This enables us to recover some results of S. Greco as well as answer one his questions regarding the canonical module of R.

Finally, we apply our study to the long standing question due to R. Berger. Around 1963, Berger conjectured that over a perfect field k, a complete local one dimensional domain R, which is a k-algebra, is regular if and only if the universally finite module of differentials of R is torsion-free. We show that this conjecture holds whenever h(-) applied to the above module satisfies a suitable upper bound in terms of the embedding dimension of R. This generalizes past work of G. Scheja and also establishes some further cases of the conjecture. We also provide an explicit algorithm to compute the h(-) in this case which in turn leads us to proving a new characterization of quasi-homogeneity of R in terms of the valuation of the trace ideal of the universally finite module of differentials.

The thesis also discusses another new approach to Berger's conjecture via extending R using the conductor ideal. This work, which was done jointly with Craig Huneke and Vivek Mukundan, establishes some further cases of the conjecture. In particular, we show that the conjecture is true whenever R is Gorenstein of embedding dimension at least 6 and the sixth power of the maximal ideal m is contained in a minimal reduction of m.