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Crossed Products and Ramification

Published online by Cambridge University Press:  22 January 2016

Susan Williamson
Susan Williamson*
Affiliation:
Regis College, Weston, Massachusetts
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Introduction. Let S be the integral closure of a complete discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, and let G denote the Galois group of the quotient field extension. Auslander and Rim have shown in [3] that the trivial crossed product Δ (1, S, G) is an hereditary order if and only if 5 is a tamely ramified extension of R. And the author has proved in [7] that if the extension S of R is tamely ramified then the crossed product Δ(f, 5, G) is a Π-principal hereditary order for each 2-cocycle f in Z2(G, U(S)). (See Section 1 for the definition of Π-principal hereditary order.) However, the author has exhibited in [8] an example of a crossed product Δ(f, S, G) which is a Π-principal hereditary order in the case when S is a wildly ramified extension of R.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 28 , October 1966 , pp. 85 - 111
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Artin, E., Nesbitt, C. and Thrall, R., Rings with Minimum Condition, Michigan (1955).Google Scholar
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[7] Williamson, S., Crossed products and hereditary orders, Nagoya Math. J., vol. 23 (1963), pp. 103–120.Google Scholar
[8] Williamson, S., Crossed products and maximal orders, Nagoya Math. J., vol. 25 (1965), pp. 165–174.CrossRefGoogle Scholar
[9] Curtis, C. and Reiner, I., Representation Theory of Finite Groups and Associative Algebras, Wiley and Sons (1962).Google Scholar
[10] Harada, M., Some criteria for hereditarity of crossed products, Osaka J. Math., vol. 1 (1964), pp. 69–80.Google Scholar