Explicit integral Galois module structure of weakly ramified extensions of local fields

Abstract:

Let $L/K$ be a finite Galois extension of complete local fields with finite residue fields and let $G=\operatorname {Gal}(L/K)$. Let $G_{1}$ and $G_{2}$ be the first and second ramification groups. Thus $L/K$ is tamely ramified when $G_{1}$ is trivial and we say that $L/K$ is weakly ramified when $G_{2}$ is trivial. Let $\mathcal {O}_{L}$ be the valuation ring of $L$ and let $\mathfrak {P}_{L}$ be its maximal ideal. We show that if $L/K$ is weakly ramified and $n \equiv 1 \bmod |G_{1}|$, then $\mathfrak {P}_{L}^{n}$ is free over the group ring $\mathcal {O}_{K}[G]$, and we construct an explicit generating element. Under the additional assumption that $L/K$ is wildly ramified, we then show that every free generator of $\mathfrak {P}_{L}$ over $\mathcal {O}_{K}[G]$ is also a free generator of $\mathcal {O}_{L}$ over its associated order in the group algebra $K[G]$. Along the way, we prove a ‘splitting lemma’ for local fields, which may be of independent interest.