# 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.