Weak crossed-product orders over valuation rings

Abstract

Let $F$ be a field, let $V$ be a valuation ring of $F$ of arbitrary Krull dimension (rank), let $K$ be a finite Galois extension of $F$ with group $G$, and let $S$ be the integral closure of $V$ in $K$. Let $f:G\times G\mapsto K\setminus \{0\}$ be a normalized two-cocycle such that $f(G\times G)\subseteq S\setminus \{0\}$, but we do not require that $f$ should take values in the group of multiplicative units of $S$. One can construct a crossed-product $V$-order $A_f={\sum }_{\sigma \in G}S{x}_{\sigma }$ with multiplication given by $x_{\sigma }s{x}_{\tau }=\sigma (s)f(\sigma ,\tau )x_{\sigma \tau }$ for $s\in S$, $\sigma ,\tau \in G$. We characterize semihereditary and Dubrovin crossed-product orders, under mild valuation-theoretic assumptions placed on the nature of the extension $K/F$.

References in corpus (1)

• On a class of hereditary crossed-product orders