Weak crossed-product orders over valuation rings

TL;DR

This paper characterizes semihereditary and Dubrovin crossed-product orders over valuation rings, extending understanding of their structure under valuation-theoretic assumptions in the context of Galois extensions.

Contribution

It provides new characterizations of semihereditary and Dubrovin crossed-product orders over valuation rings without requiring cocycle values to be units.

Findings

Characterization of semihereditary crossed-product orders

Characterization of Dubrovin crossed-product orders

Results depend on valuation-theoretic properties of the extension

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 $A_f=\sum_{\sigma\in G}Sx_{\sigma}$ with multiplication given by $x_{\sigma}sx_{\tau}=\sigma(s)f(\sigma,\tau)x_{\sigma\tau}$ 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 $K/F$.