Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-15T04:19:19.415Z Has data issue: false hasContentIssue false
  • English
  • Français

Galois Extensions as Modules over the Group Ring

Published online by Cambridge University Press:  20 November 2018

Gerald Garfinkel  and
Morris Orzech
Gerald Garfinkel
Affiliation:
University of Illinois, Urbana, Illinois
Morris Orzech
Affiliation:
University of Illinois, Urbana, Illinois and Queen's University, Kingston, Ontario
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.

Suppose that R is a commutative ring and G is a finite abelian group. In § 2 we review the definition of E(R, G) (T(R, G)), the group of all (commutative) Galois extensions S of R with Galois group G. We discuss the properties of these groups as functors of G and give an example which exhibits some of the pathological properties of the functor E(R, – ). In § 3 we display a homomorphism from E(R, G) to Pic (R(G)); we use this homomorphism to prove that if S is commutative, G has exponent m, and R(G) has Serre dimension 0 or 1, then a direct sum of m copies of S is isomorphic as a G-module to a direct sum of m copies of R(G). (This result is related to [5, Theorem 4.2], where it is shown that if S is a free R-module and G is any finite group with n elements, then Sn is isomorphic to R(G)n as G-modules.) We also give some examples of Galois extensions without normal bases.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 242 - 248
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Bass, H., K-theory and stable algebras, Inst. Hautes Etudes Sci. Publ. Math. No. 22 (1964), 5–60.Google Scholar
2. Bourbaki, N., Eléments de mathématique, Fasc. XXVII, Algèbre commutative, Chapitre 1: Modules plats, Chapitre 2: Localization, Actualités Sci. Indust., No. 1290 (Hermann, Paris, 1961).Google Scholar
3. Chase, S. U. and Alex, Rosenberg, Amitsur cohomology and the Brauer group, Mem. Amer. Math. Soc. No. 52 (1965), 34–79.Google Scholar
4. Chase, S. U. and Alex, Rosenberg, A theorem of Harrison, Kummer theory, and Galois algebras, Nagoya Math. J. 27 (1966), 663–685.Google Scholar
5. Chase, S. U., Harrison, D. K., and Alex, Rosenberg, Galois theory and Galois cohomology of commutative rings, Mem. Amer. Math. Soc. No. 52 (1965), 15–33.Google Scholar
6. Harrison, D. K., Abelian extensions of commutative rings, Mem. Amer. Math. Soc. No. 52 (1965), 1–14.Google Scholar
7. Orzech, M., A cohomological description of abelian Galois extensions, Trans. Amer. Math. Soc. 137 (1969), 481–499.Google Scholar
8. Spanier, E. H., Algebraic topology (McGraw-Hill, New York, 1966).Google Scholar
9. Takeuchi, Y., On Galois extensions over commutative rings, Osaka J. Math. 2 (1965), 137–145.Google Scholar