Crossed-products orders over discrete valuation rings
The generalized cohomology theory of \textit{D. E. Haile}, \textit{R. G. Larson} and \textit{M. E. Sweedler} [Am. J. Math. 105, 689-814 (1983; Zbl 0523.13005)] is applied to the construction of crossed-product orders analogous to the classical crossed-product algebras. Let R be a discrete valuation ring with field of fractions F, K/F a finite Galois extension with group G and S the integral closure of R in K. S/R is assumed to be unramified. For any 2-cocycle \(f: G\times G\to S^*\) the author forms a crossed-product order \(A_ f=\coprod_{\sigma \in G}Sx_{\sigma}\) with the usual rules of multiplication which is an order in the crossed- product algebra \(\Sigma_ f=\coprod_{\sigma \in G}Kx_{\sigma}.\) From the author's introduction: ''In the first section we derive some basic properties of the cocycles and the orders. We show how to associate a finite graph to each cocycle. One of the themes of the paper is the relationship between properties of this graph and the structure of the order. Also in this section we show that if the residue field is perfect, then every maximal order is equivalent to a crossed-product order. In Section 2 we assume that S is a discrete valuation ring. In this case the crossed-product orders are primary and there is a simple characterization of those cocycles which give rise to maximal orders. Also in this section we show how to determine the ideals in the order and give necessary and sufficient conditions for two orders \(A_{f_ 1}\) and \(A_{f_ 2}\) to be isomorphic as R-algebras. In Section 3 we take up the general case (S/R unramified but not necessarily local). Let M be a maximal ideal of S with decomposition group \(D_ M\). If \(f: G\times G\to S\) is a cocycle, then \(A_ f\) is primary if and only if there are coset representatives \(g_ 1,...,g_ r\) of \(D_ M\) in G with \(f(g_ i,g_ i^{-1})\not\in M\) for all i. If \(f_ M: D_ M\times D_ M\to S\) denotes the restriction of f, then we can form the new crossed-product order \(A_ f\). If \(A_ f\) is primary, then there is a one-to-one product preserving correspondence between the ideals of \(A_ f\) and the ideals of \(A_{f_ M}\). In particular, in the case where \(A_ f\) is primary, we show that \(A_ f\) is maximal if and only if \(A_{f_ M}\) is maximal. Along the way we obtain results on the relationship between the graph of f and the graphs of the cocycles \(f_ M\), as M varies through the maximal ideals of S. We also show how to compute the ideals of primary orders and end the section with a determination of when two maximal crossed-product orders are R-isomorphic. In the final section, examples are given of the various definitions and results of the preceding sections.''
scientific article; zbMATH DE number 3985440
scientific article