Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-08T10:28:20.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Crossed Products and Hereditary Orders

Published online by Cambridge University Press:  22 January 2016

Susan Williamson
Susan Williamson*
Affiliation:
Brandeis University, Waltham, Massachusetts
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let S be the integral closure of a discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, and denote the Galois group of the quotient field extension by G. It has been proved by Auslander and Rim in [4] that the trivial crossed product Δ(l, S, G) is an hereditary order for tamely ramified extensions S of R and that Δ(l, S, G) is a maximal order if and only if S is an unramified extension of R. The purpose of this paper is to study the crossed product Δ(f, S, G) where [f] is any element of H2(G, U(S)) and S is a tamely ramified extension of R with multiplicative group of units U(S).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 23 , December 1963 , pp. 103 - 120
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1963

References

[1] Artin, E., C. Nesbitt and R. Thrall, 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. 367–409.Google Scholar
[3] Auslander, M. and Goldman, O., Maximal orders, Trans, Amer. Math. Soc. vol. 97 (1960), pp. 1–24.Google Scholar
[4] Auslander, M. and Rim, D. S., Ramification Index and Multiplicity, 111. J. of Math., Vol. 7 (1963) pp. 566–581.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. 273–290.Google Scholar
[7] Northcott, D. G., An Introduction to Homological Algebra, Cambridge University Press (1960).CrossRefGoogle 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