Thursday 25 Jul 2013Some counterexamples to a naive generalization of Noether's theorem to nonclassical Hopf-Galois structures

Dr Paul Truman - University of Keele

Harrison 203 15:00-16:00

Let $ L/K $ be a finite Galois extension of $ p $-adic fields with group $ G $, and suppose that $ L/K $ is at most tamely ramified. By Noether's theorem, the valuation ring $ {\cal O}_{L} $ is a free module of rank one over the integral group ring $ {\cal O}_{K}[G] $. We might wonder whether similar results hold for nonclassical Hopf-Galois structures on the extension. If $ H $ is a Hopf algebra giving such a structure, then $ H $ has the form $ L[N]^{G} $ for some group $ N $ of the same cardinality as $ G $. In previous work we identified a number of cases in which {\cal O}_{L} $ is free of rank one over the order $ {\cal O}_{L}[N]^{G} $. In this talk we show that this is usually not the case, which leads us to wonder what conditions are necessary or sufficient for it to occur.

Add to calendar

Add to calendar (.ics)