Galois module structure of the square root of the inverse different over maximal orders

Abstract

Let $K$ be a number field with ring of integers ${\mathcal{O}}_K$ and let $G$ be a \mbox{finite group of} odd order. Given a $G$-Galois $K$-algebra $K_h$, let $A_h$ be the square root of the inverse different of $K_h/K$, which exists by Hilbert's formula. If $K_h/K$ is weakly ramified, then $A_h$ is locally free over ${\mathcal{O}}_{K}G$ by a result of B. Erez, in which case it determines a class in the locally free class group $\mbox{Cl}(\mathcal{O}_KG)$ of ${\mathcal{O}}_{K}G$. Such a class in $\mbox{Cl}(\mathcal{O}_KG)$ is said to be $A$-realizable, and tame $A$-realizable if $K_h/K$ is tame. Let $\mathcal{A}({\mathcal{O}}_{K}G)$ and ${\mathcal{A}}^t({\mathcal{O}}_{K}G)$ denote the sets of all $A$-realizable classes and tame $A$-realizable classes, respectively. For $G$ abelian, we will show that the two sets $\mathcal{A}({\mathcal{O}}_{K}G)$ and ${\mathcal{A}}^t({\mathcal{O}}_{K}G)$ are equal when extended scalars to the maximal order in $KG$.

We improved one of the results in arxiv.org/abs/1407.4175

References in corpus (1)

• On the Galois module structure of the square root of the inverse different in abelian extensions