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Patching over fields

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Abstract

We develop a new form of patching that is both far-reaching and more elementary than the previous versions that have been used in inverse Galois theory for function fields of curves. A key point of our approach is to work with fields and vector spaces, rather than rings and modules. After presenting a self-contained development of this form of patching, we obtain applications to other structures such as Brauer groups and differential modules.

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References

  1. N. Bourbaki, Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass., 1972.

    MATH  Google Scholar 

  2. H. Cartan, Sur les matrices holomorphes de n variables complexes, Journal de Mathématiques Pures et Appliquées 19 (1940), 1–26.

    MATH  MathSciNet  Google Scholar 

  3. M. D. Fried and M. Jarden, Field Arithmetic, second edition, Springer, Berlin, 2005.

    MATH  Google Scholar 

  4. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New York, 1978.

    MATH  Google Scholar 

  5. A. Grothendieck, Élements de Géométrie Algébrique (EGA) III, 1e partie, Publications Mathématiques Institut de Hautes Études Scientifiques 11 (1961).

  6. D. Haran and M. Jarden, Regular split embedding problems over complete valued fields, Forum Mathematicum 10 (1998), 329–351.

    Article  MATH  MathSciNet  Google Scholar 

  7. D. Haran and M. Jarden, Regular split embedding problems over function fields of one variable over ample fields, Journal of Algebra 208 (1998), 147–164.

    Article  MATH  MathSciNet  Google Scholar 

  8. D. Haran and H. Völklein, Galois groups over complete valued fields, Israel Journal of Mathematics 93 (1996), 9–27.

    Article  MATH  MathSciNet  Google Scholar 

  9. D. Harbater, Convergent arithmetic power series, American Journal of Mathematics 106 (1984), 801–846.

    Article  MATH  MathSciNet  Google Scholar 

  10. D. Harbater, Galois coverings of the arithmetic line, in Number Theory (New York, 1984–1985) (D. V. Chudnovsky, and G. V. Chudnovsky, eds.), Volume 1240 of Lecture Notes in Mathematics, Springer, Berlin, 1987, pp. 165–195.

    Google Scholar 

  11. D. Harbater, Formal patching and adding branch points, American Journal of Mathematics 115 (1993), 487–508.

    Article  MATH  MathSciNet  Google Scholar 

  12. D. Harbater, Fundamental groups and embedding problems in characteristic p, in Recent Developments in the Inverse Galois Problem (M. Fried, ed.) Volume 186 of Contemporary Mathematics Series, American Math. Society, Providence, RI, 1995, pp. 353–369.

    Google Scholar 

  13. D. Harbater, Patching and Galois theory, in Galois Groups and Fundamental Groups (L. Schneps, ed.), MSRI Publications Series, Vol. 41, Cambridge University Press, Cambridge, 2003, pp. 313–424.

    Google Scholar 

  14. D. Harbater, Patching over fields (joint work with Julia Hartmann), in Arithmetic and Differential Galois Groups, Volume 4, no. 2 of Oberwolfach reports, European Mathematical Society, Zürich, 2007.

    Google Scholar 

  15. D. Harbater and J. Hartmann, Patching and differential Galois groups, 2007, preprint.

  16. D. Harbater, J. Hartmann and D. Krashen, Patching subfields of division algebras, 2007, submitted, available at arXiv:0904.1594.

  17. D. Harbater and K. F. Stevenson, Patching and thickening problems, Journal of Algebra 212 (1999), 272–304.

    Article  MATH  MathSciNet  Google Scholar 

  18. J. Hartmann, Patching and differential Galois groups (joint work with David Harbater), in Arithmetic and Differential Galois Groups, Volume 4, no. 2 of Oberwolfach reports, European Mathematical Society, Zürich, 2007.

    Google Scholar 

  19. R. Hartshorne, Algebraic Geometry, Springer, Berlin, 1977.

    MATH  Google Scholar 

  20. I. N. Herstein, Noncommutative Rings, in The Carus Mathematical Monographs, No. 15, Published by The Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York 1968.

    MATH  Google Scholar 

  21. M. Jarden, The inverse Galois problem over formal power series fields, Israel Journal of Mathematics 85 (1994), 263–275.

    Article  MATH  MathSciNet  Google Scholar 

  22. Q. Liu, Tout groupe fini est un groupe de Galois sur ℚ p (T), d’après Harbater, in Recent Developments in the Inverse Galois Problem (M. Fried, ed.) Volume 186 of Contemporary Mathematics Series, American Math. Society, Providence, RI, 1995, pp. 261–265.

    Google Scholar 

  23. B. H. Matzat and M. van der Put, Constructive differential Galois theory, in Galois Groups and Fundamental Groups (L. Schneps, ed.), Volume 41 of MSRI Publications Series, Cambridge University Press, Cambridge, 2003, pp. 425–467.

    Google Scholar 

  24. D. Mumford, The Red Book of Varieties and Schemes, Volume 1358 of Lecture Notes in Mathematics, Springer, Berlin, 1988 and 1999.

    MATH  Google Scholar 

  25. R. Pierce, Associative Algebras, Springer, Berlin, 1982.

    MATH  Google Scholar 

  26. F. Pop, Étale Galois covers of smooth affine curves, Inventiones Mathematicae 120 (1995), 555–578.

    Article  MATH  MathSciNet  Google Scholar 

  27. R. Pries, Construction of covers with formal and rigid geometry, in Courbes Semi-stables et Groupe Fondamental en Géométrie Algébrique (J. B. Bost, F. Loeser and M. Raynaud, eds.),Volume 187 of Progress in Mathematics, Birkhäuser, Basel, 2000, pp. 157–167.

    Google Scholar 

  28. M. Raynaud, Revêtements de la droite affine en caractéristique p > 0 et conjecture d’Abhyankar, Inventiones Mathematicae 116 (1994), 425–462.

    Article  MATH  MathSciNet  Google Scholar 

  29. J. P. Serre, Géométrie algébrique et géométrie analytique, Annales de l’Institut Fourier 6 (1956), 1–42.

    MathSciNet  Google Scholar 

  30. J. P. Serre, Algebraic Groups and Class Fields, Springer, Berlin, 1988.

    MATH  Google Scholar 

  31. H. Völklein, Groups as Galois Groups, Cambridge University Press, Cambridge, 1996.

    Book  MATH  Google Scholar 

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Correspondence to David Harbater.

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Supported in part by NSF Grant DMS-0500118.

Supported by the German National Science Foundation (DFG).

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Harbater, D., Hartmann, J. Patching over fields. Isr. J. Math. 176, 61–107 (2010). https://doi.org/10.1007/s11856-010-0021-1

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  • DOI: https://doi.org/10.1007/s11856-010-0021-1

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