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Crossed Products and Maximal Orders

Published online by Cambridge University Press:  22 January 2016

Susan Williamson
Susan Williamson*
Affiliation:
Harvard University, Cardinal Gushing College
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Let I’ be a maximal order over a complete discrete rank one valuation ring R in a central simple algebra over the quotient field of R. The purpose of this paper is to determine necessary and sufficient conditions for I’ to be equivalent to a crossed product over a tamely ramified extension of R.

It is a classical result that every central simple algebra over a field k is equivalent to a crossed product over a Galois extension of k. Furthermore, it has been proved by Auslander and Goldman in [2] that every central separable algebra over a local ring is equivalent to a crossed product over an unramified extension.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 25 , 1965 , pp. 165 - 174
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1965

References

[1] Artin, E., Nesbitt, C. and Thrall, C., Rings with Minimum Condition, Michigan, (1955).Google Scholar
[2] Auslander, M. and Goldman, O., The Brauer group of a commutative ring, Trans. Amer. Math. Soc. Vol. 97 (1960), pp. 367–367.Google Scholar
[3] Auslander, M. and Goldman, O., Maximal orders, Trans. Amer. Math. Soc. Vol. 97 (1960), pp. 1–1.Google Scholar
[4] Auslander, M. and Rim, D. S., Ramification Index and Multiplicity, 111. J. of Math. Vol.7 (1963), pp. 566–566.Google Scholar
[5] Cartan, H. and Eilenberg, S., Homological Algebra, Princeton, (1956).Google Scholar
[6] Harada, M., Hereditary orders, Trans. Amer. Math. Soc. Vol. 107 (1963), pp. 273–273.Google Scholar
[7] Northcott, D. G., An Introduction to Homological Algebra, Cambridge University Press, (1960).Google Scholar
[8] Zariski, O. and Samuel, P., Commutative Algebra, Vol. I, Van Nostrand, (1958).Google Scholar
[9] Zariski, O. and Samuel, P., Commutative Algebra, Vol. II, Van Nostrand (1960).Google Scholar
[10] Williamson, S., Crossed Products and Hereditary Orders, Nagoya Math. J. Vol.23 (1963), pp. 103–103.Google Scholar
[11] Bourbaki, N., Elements de Mathématique, Livre II, Algèbre, Chap. V, Paris, Hermann, (1950).Google Scholar