On a class of semihereditary crossed-product orders

TL;DR

This paper characterizes when certain crossed-product orders over valuation rings are semihereditary, linking algebraic properties to cocycle conditions and the structure of the valuation ring.

Contribution

It provides necessary and sufficient conditions for semiheredity of crossed-product orders over valuation rings, extending understanding beyond unramified and defectless cases.

Findings

A crossed-product order is semihereditary iff cocycle values avoid squares of maximal ideals.

Semihereditary orders are Azumaya if the Jacobson radical is not principal.

The characterization applies to valuation rings of arbitrary Krull dimension.

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 $V$-algebra $A_f=\sum_{\sigma\in G}Sx_{\sigma}$ in a natural way, which is a $V$-order in the crossed-product $F$-algebra $(K/F,G,f)$. If $V$ is unramified and defectless in $K$, we show that $A_f$ is semihereditary if and only if for all $\sigma,\tau\in G$ and every maximal ideal $M$ of $S$, $f(\sigma,\tau)\not\in M^2$. If in addition $J(V)$ is not a principal ideal of $V$, then $A_f$ is semihereditary if and only if it is an Azumaya algebra over $V$.