Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-24T00:04:01.429Z Has data issue: false hasContentIssue false

On Ramification Theory in Projective Orders

Published online by Cambridge University Press:  22 January 2016

Shizuo Endo*
Affiliation:
Tokyo University of Education
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.

The ramification theory in commutative rings, as a generalization of the classical one in maximal orders over a Dedekind domain, was established in [1], [13], [15] and etc.. For non-commutative algebras this was also studied in [3], [4], [9], [18] and etc.. However, the different theorem, which is a central part of ramification theory, has not been given in those, except for some special cases (cf. [12], [18]). The main object of this paper is to give the discriminant theorem and the different theorem for projective orders in a (non-commutative) separable algebra, in the most general form.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1969

References

[1] Auslander, M. and Buchsbaum, D.A., On ramification theory in Noetherian rings, Amer. J. Math., 81 (1959), 749765.CrossRefGoogle Scholar
[2] Auslander, M. and Goldman, O., Maximal orders, Trans. Amer. Math. Soc, 97 (1960), 124.Google Scholar
[3] Auslander, M. and Goldman, O., The Brauer group of a commutative ring, Trans. Amer. Math. Soc, 97 (1960), 367409.Google Scholar
[4] DeMeyer, F. R., The trace map and separable algebra, Osaka J. Math., 3 (1966), 711.Google Scholar
[5] Deuring, M., Algebren, Springer, Berlin, 1935.Google Scholar
[6] Endo, S., Completely faithful modules and quasi-Frobenius algebras, J. Math. Soc. Japan, 19 (1967), 437456.Google Scholar
[7] Endo, S. and Watanabe, Y., On separable algebras over a commutative ring, Osaka J. Math., 4 (1967), 233242.Google Scholar
[8] Endo, S. and Watanabe, Y., The centers of semi-simple algebras over a commutative ring, II, to appear.Google Scholar
[9] Fossum, R., The Noetherian different of projective orders, J. reine angew. Math., 224 (1966), 209218.Google Scholar
[10] Harada, M., Hereditary orders, Trans. Amer. Math. Soc, 107 (1963), 273290.Google Scholar
[11] Harada, M., Structure of hereditary orders over local rings, J. Math. Osaka City Univ., 14 (1963), 122.Google Scholar
[12] Harada, M., Multiplicative ideal theory in hereditary orders, J. Math. Osaka City Univ., 14 (1963), 83106.Google Scholar
[13] Kunz, E., Die Primidealteiler der Differenten in allgemeinen Ringen, J. reine angew. Math., 204 (1960), 165182.Google Scholar
[14] Nagata, M., Local rings, Interscience Publ., New York, 1962.Google Scholar
[15] Nakai, Y., On the theory of differentials in commutative rings, J. Math. Soc. Japan, 13 (1961), 6384.Google Scholar
[16] Noether, E., Ideal-Differentiation und Differente, J. reine angew. Math., 188 (1950), 120.CrossRefGoogle Scholar
[17] Schilling, O. F. G., The theory of valuations, Publ. A.M.S., 1950.CrossRefGoogle Scholar
[18] Watanabe, Y., The Dedekind different and the homological different, Osaka J. Math., 4 (1967), 227231.Google Scholar