arXiv subjects

John S. Kauta

Publications and source records attributed to John S. Kauta.

3 recordsLinked to original sources

arXiv2014

Weak crossed-product orders over valuation rings

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 $A_f=\sum_{σ\in G}Sx_σ$ with multiplication given by $x_σsx_τ=σ(s)f(σ,τ)x_{στ}$ for $s\in S$, $σ,τ\in G$. We characterize semihereditary and Dubrovin crossed-product orders, under mild valuation-theoretic assumptions placed on the nature of the extension $K/F$.

math.RA↗

arXiv2014

On a class of semihereditary crossed-product orders

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 $A_f=\sum_{σ\in G}Sx_σ$ 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 $σ,τ\in G$ and every maximal ideal $M$ of $S$, $f(σ,τ)\not\in M^2$. 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 $V$.

math.RA↗

arXiv2012

On a class of hereditary crossed-product orders

In this brief note, we revisit a class of crossed-product orders over discrete valuation rings introduced by D. E. Haile. We give simple but useful criteria, which involve only the two-cocycle associated with a given crossed-product order, for determining whether such an order is a hereditary order or a maximal order.

math.RA↗