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arXiv · 0908.4562

Integral Galois Module Structure for Elementary Abelian Extensions with a Galois Scaffold

Nigel P. Byott·G. Griffith Elder
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Abstract

This paper justifies an assertion in (Elder, Proc AMS 137 (2009), no 4, 1193--1203) that Galois scaffolds make the questions of Galois module structure tractable. Let $k$ be a perfect field of characteristic $p$ and let $K=k((T))$. For the class of characteristic $p$ elementary abelian $p$-extensions $L/K$ with Galois scaffolds described in mentioned paper, we give a necessary and sufficient condition for the valuation ring $\mathfrak{O}_L$ to be free over its associated order $\mathcal{A}_{L/K}$ in $K[\Gal(L/K)]$. Interestingly, this condition agrees with the condition found by Y. Miyata, concerning a class of cyclic Kummer extensions in characteristic zero.

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Nigel P. Byott, G. Griffith Elder. 2009-08-31. Integral Galois Module Structure for Elementary Abelian Extensions with a Galois Scaffold. https://arxiv.org/abs/0908.4562

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