Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T23:20:34.013Z Has data issue: false hasContentIssue false
  • English
  • Français

An Exact Sequence Associated with a Generalized Crossed Product

Published online by Cambridge University Press:  22 January 2016

Yôichi Miyashita
Yôichi Miyashita*
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 purpose of this paper is to generalize the seven terms exact sequence given by Chase, Harrison and Rosenberg [8]. Our work was motivated by Kanzaki [16] and, of course, [8], [9]. The main theorem holds for any generalized crossed product, which is a more general one than that in Kanzaki [16]. In §1, we define a group P(A/B) for any ring extension A/B, and prove some preliminary exact sequences. In §2, we fix a group homomorphism J from a group G to the group of all invertible two-sided B-submodules of A.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 49 , March 1973 , pp. 21 - 51
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1973

References

[1] Auslander, M. and Goldman, O.: The Brauer group of a commutative ring, Trans. Amer. Math. Soc, 19 (1960), 367–409.Google Scholar
[2] Azumaya, G.: Algebraic theory of simple rings (in Japanese), Kawade Syobô, Tokyo, 1952.Google Scholar
[3] Azumaya, G.: Maximally central algebras, Nagoya Math. J., 2 (1951), 119–150.Google Scholar
[4] Azumaya, G.: Completely faithful modules and self injective rings, Nagoya Math. J., 27 (1966), 697–708.Google Scholar
[5] Bass, H.: The Morita Theorems, Lecture note at Univ. of Oregon, 1962.Google Scholar
[6] Bass, H.: Lectures on topics in algebraic K-theory, Tata Institute of Fundamental Research, Bombay, 1967.Google Scholar
[7] Bass, H.: Algebraic K-theory, Benjamin, 1968.Google Scholar
[8] Chase, S. U., Harrison, D. K. and Rosenberg, A.: Galois Theory and Galois cohomology of commutative rings, Mem. Amer. Math. Soc, 52 (1965).Google Scholar
[9] Chase, S. U. and Rosenberg, A.: Amitzur complex and Brauer group, Mem. Amer. Math. Soc, 52 (1965).Google Scholar
[10] DeMeyer, F. R.: Some note on the general Galois theory of rings, Osaka J. Math., 2 (1965), 117–127.Google Scholar
[11] DeMeyer, F. R. and Ingraham, E.: Separable algebras over commutative rings, Springer, 1971.Google Scholar
[12] Harrison, D. K.: Abelian extensions of commutative rings, Mem. Amer. Math. Soc, 52 (1965).Google Scholar
[13] Hirata, K.: Some types of separable extensions of rings, Nagoya Math. J., 33 (1968), 108–115.Google Scholar
[14] Hirata, K.: Separable extensions and centralizers of rings, Nagoya Math. J., 35 (1969), 31–45.CrossRefGoogle Scholar
[15] Kanzaki, T.: On Galois algebras over a commutative ring, Osaka J. Math., 2 (1965), 309–317.Google Scholar
[16] Kanzaki, T.: On generalized crossed product and Brauer group, Osaka J. Math., 5 (1968), 175–188.Google Scholar
[17] Miyashita, Y.: Finite outer Galois theory of non-commutative rings, J. Fac Sci. Hokkaido Univ., Ser. I, 19 (1966), 114–134.Google Scholar
[18] Miyashita, Y.: Galois extensions and crossed products, J. Fac Sci. Hokkaido Univ., Ser. I, 20 (1968), 122–134.Google Scholar
[19] Miyashita, Y.: On Galois extensions and crossed products, J. Fac. Sci. Hokkaido Univ., Ser. I, 21 (1970), 97–121.Google Scholar
[20] Morita, K.: Duality for modules and its application to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyôiku Daigaku, 6 (1958), 83–142.Google Scholar