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Thursday 30 Oct 2014Hopf orders in elementary abelian group rings

G Griffith Elder - University of Nebraska at Omaha

Harrison 103 15:00-16:00

Let $p$ be a prime number, $R$ be a discrete valuation ring whose prime ideal $P$ contains $p$, $K$ be the field of fractions for $R$, and so that we can focus on the difficulties when only one prime is involved, let our group $G$ be a $p$-group. The question of $R$-Hopf orders in $KG$ begins with Tate and Oort in 1970, with important contributions by Greither, Byott, Underwood and Childs in the past two and a half decades. Still the classifications for $G$ cyclic or elementary abelian of order $p^3$ remain incomplete. In this talk I will describe a characteristic independent approach to first construct Hopf orders in $KG$, and then hopefully classify them. I will describe how this approach has emerged out of local Galois module theory, and why a characteristic independent approach ought to be considered natural. I will then conclude with recent results of this approach for the case where $G$ is elementary abelian of order $p^n$.

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