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Galois Extensions as Modules over the Group Ring

  • Gerald Garfinkel (a1) and Morris Orzech (a2)

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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.

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References

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1. Bass, H., K-theory and stable algebras, Inst. Hautes Etudes Sci. Publ. Math. No. 22 (1964), 560.
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).
3. Chase, S. U. and Alex, Rosenberg, Amitsur cohomology and the Brauer group, Mem. Amer. Math. Soc. No. 52 (1965), 3479.
4. Chase, S. U. and Alex, Rosenberg, A theorem of Harrison, Kummer theory, and Galois algebras, Nagoya Math. J. 27 (1966), 663685.
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), 1533.
6. Harrison, D. K., Abelian extensions of commutative rings, Mem. Amer. Math. Soc. No. 52 (1965), 114.
7. Orzech, M., A cohomological description of abelian Galois extensions, Trans. Amer. Math. Soc. 137 (1969), 481499.
8. Spanier, E. H., Algebraic topology (McGraw-Hill, New York, 1966).
9. Takeuchi, Y., On Galois extensions over commutative rings, Osaka J. Math. 2 (1965), 137145.
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Galois Extensions as Modules over the Group Ring

  • Gerald Garfinkel (a1) and Morris Orzech (a2)

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