The problem of determining when a (classical) crossed product $T=S^f*G$ of a finite group $G$ over a discrete valuation ring $S$ is a maximal order, was answered in the 1960's for the case where $S$ is tamely ramified over the subring of invariants $S^G$. The answer was given in terms of the conductor subgroup (with respect to $f$) of the inertia. In this paper we solve this problem in general when $S/S^G$ is residually separable. We show that the maximal order property entails a restrictive structure on the sub-crossed product graded by the inertia subgroup. In particular, the inertia is abelian. Using this structure, one is able to extend the notion of the conductor. As in the tame case, the order of the conductor is equal to the number of maximal two sided ideals of $T$ and hence to the number of maximal orders containing $T$ in its quotient ring. Consequently, $T$ is a maximal order if and only if the conductor subgroup is trivial.
Degeneracy and decomposability in abelian crossed products
本文研究交换交叉积的退化性与可分解性之间的关系。特别地，当$p$为奇素数时，我们构造了一个指数为$p$，指数为$p^2$的不可分解交换叉积除代数.我们构造的代数在Amitsur和Saltman意义下是一般的，并具有其基础交换交叉积是由非退化矩阵定义的可分解除代数的性质.这个代数给出了一个不可分解的一般阿贝尔交叉积的例子，它被证明是不可分解的，而不使用相应的Severi-Brauer簇的Chow群中的挠，正如[Karpenko，Cowimension 2cycles on Severi-Brauer varieites（1998）]和[McKinnie，Indecomposable $p$ -algebras and Galois subfields in generic abelian crossed products（2008）]中所需要的那样。它还给出了一个例子，Brauer类是在Tabel的Dec群关于一个交换极大子域，但不是在Dec群关于另一个。
证明了每一个dp-极小整环$R$是局部环，并且对于$R$的每一个非极大素理想$\mathfrak p $，局部化$R_{\mathfrak p }$是赋值环，$\mathfrak{p}R_{\mathfrak{p}}=\mathfrak{p}$ .进一步证明了dp-极小整环是赋值环当且仅当它的剩余域是无限的或它的剩余域是有限的且它的极大理想是主的.
