Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-19T11:31:40.859Z Has data issue: false hasContentIssue false
  • English
  • Français

Groups, Coverings and Galois Theory

Published online by Cambridge University Press:  20 November 2018

Vagn Lundsgaard Hansen  and
Peter Petersen V
Vagn Lundsgaard Hansen
Affiliation:
Mathematical Institute, The Technical University of Denmark, Building 303, DK-2800 Lyngby, Denmark
Peter Petersen V
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA 90024, U. S. A.
Rights & Permissions [Opens in a new window]

Abstract

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.

Finite extensions of complex commutative Banach algebras are naturally related to corresponding finite covering maps between the carrier spaces for the algebras. In the case of function rings, the finite extensions are induced by the corresponding finite covering maps, and the topological properties of the coverings are strongly reflected in the algebraic properties of the extensions and conversely. Of particular interest to us is the class of finite covering maps for which the induced extensions of function rings admit primitive generators. This is exactly the class of polynomial covering maps and the extensions are algebraic extensions defined by the underlying Weierstrass polynomials.

The purpose of this paper is to develop a suitable Galois theory for finite extensions of function rings induced by finite covering maps and to apply it in the case of Weierstrass polynomials and polynomial covering maps.

Keywords

57M1213B2546J10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 6 , 01 December 1991 , pp. 1281 - 1293
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Arnold, V.I., Topological invariants of algebraic functions, II, Functional Anal, i Prilozen. (2) 4(1970), 1–9.and Functional Analysis and its Applications (2)4(1970), 91–98.Google Scholar
2. Childs, L.N., On covering spaces and Galois extensions, Pacific J. Math. 37(1971), 29–33.Google Scholar
3. DeMeyer, F.R. and Ingraham, E.C., Separable algebras over commutative rings. Springer Lecture Notes in Mathematics, 81, 1971.Google Scholar
4. Duchamp, T. and Hain, R.M., Primitive elements in rings of holomorphicfunctions, J. Reine Angew. Math. 346(1984), 199–220.Google Scholar
5. Gillman, L., Henriksen, M. and Jerison, M., On a theorem of Gelfand-Kolmogoroff concerning maximal ideals in rings of continuous functions, Proc. Amer. Math. Soc. 5(1954), 447–455.Google Scholar
6. Gorin, E.A. and Lin, V. Ja., Algebraic equations with continuous coefficients, and certain questions of the algebraic theory of braids, Mat. Sb.78(120)(1969), 579–610. and Math. Sbornik 7(1969), 569–596.Google Scholar
7. Hansen, V.L., Coverings defined by Weierstrass polynomials, J. Reine Angew. Math. 314(1980), 29–39.Google Scholar
8. Hansen, V.L., Polynomial covering spaces and homomorphisms into the braid groups, Pacific J. Math. 81(1979), 399–410.Google Scholar
9. Hansen, V.L., Algebra and topology of Weierstrass polynomials, Expo. Math. 5(1987), 261–21.A.Google Scholar
10. Hansen, V.L., The characteristic algebra of a polynomial covering map, Math. Scand. 64(1989), 219–225.Google Scholar
11. Hansen, V.L., Polynomial covering maps. In: Braids (Libgober, A. and Birman, J.S. eds.), Contemporary Math. 78(1988), 229–243.Google Scholar
12. Hansen, V.L., Braids and coverings—selected topics. London Math. Soc. Student Texts 18, Cambridge University Press, 1989.Google Scholar
13. Lang, S., Algebra. Addison-Wesley, 1965.Google Scholar
14. Magid, A.R., Algebraically separable extensions ofBanach alagebras, Michigan Math. J. 21(1974), 137–143.Google Scholar
15. Massey, W.S., Algebraic topology: an introduction. 4th ed., Graduate Texts in Mathematics 56, Springer- Verlag, 1977.Google Scholar
16. Møller, J.M., Equivariant embeddings of principal Zn-bundles into complex vector bundles, Topology Appl. 16(1983), 279–286.Google Scholar
17. Petersen, P., V, Fatness of covers. Thesis, University of Maryland, College Park, 1987.Google Scholar
18. Petersen, P., Fatness of covers, J. Reine Angew. Math. 403(1990), 154–165.Google Scholar
19. Wajnryb, B., Projective and separable extensions of rings of continuous functions, Bull. Acad. Pol. Sci. Sér. Sci. Math. Astr. Phys. 17(1969), 269–271.Google Scholar
20. Zame, W.R., Covering spaces and the Galois theory of commutative Banach algebras, J. Functional Analysis 55(1984), 151–175.Google Scholar