On a class of semihereditary crossed-product orders

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 $V$.