event
Thursday 25 Jul 2013: Some 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.
