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The differential Galois theory of strongly normal extensions


Author: Jerald J. Kovacic
Journal: Trans. Amer. Math. Soc. 355 (2003), 4475-4522
MSC (2000): Primary 12H05, 12F10; Secondary 14A15, 14L15
DOI: https://doi.org/10.1090/S0002-9947-03-03306-3
Published electronically: July 2, 2003
MathSciNet review: 1990759
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Abstract: Differential Galois theory, the theory of strongly normal extensions, has unfortunately languished. This may be due to its reliance on Kolchin's elegant, but not widely adopted, axiomatization of the theory of algebraic groups. This paper attempts to revive the theory using a differential scheme in place of those axioms. We also avoid using a universal differential field, instead relying on a certain tensor product.

We identify automorphisms of a strongly normal extension with maximal differential ideals of this tensor product, thus identifying the Galois group with the closed points of an affine differential scheme. Moreover, the tensor product has a natural coring structure which translates into the Galois group operation: composition of automorphisms.

This affine differential scheme splits, i.e. is obtained by base extension from a (not differential, not necessarily affine) group scheme. As a consequence, the Galois group is canonically isomorphic to the closed, or rational, points of a group scheme defined over constants. We obtain the fundamental theorem of differential Galois theory, giving a bijective correspondence between subgroup schemes and intermediate differential fields.

On the way to this result we study certain aspects of differential algebraic geometry, e.g. closed immersions, products, local ringed space of constants, and split differential schemes.


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  • 1. Atiyah, M. F.; MacDonald, I. G. Introduction to commutative algebra. Addison-Wesley, Reading, Mass. 1969, MR 39:4129.
  • 2. Bourbaki, Nicolas. Commutative algebra. Chapters 1-7. Translated from the French, Addison-Wesley, Reading, Mass., 1972; latest reprint, Springer-Verlag, Berlin, 1998, MR 50:12997; MR 2001g:13001.
  • 3. Buium, Alexandru. Differential function fields and moduli of algebraic varieties. Lecture Notes in Mathematics, 1226. Springer-Verlag, Berlin, 1986, MR 88e:14010.
  • 4. Cohn, Paul Moritz. Skew field constructions. London Mathematical Society Lecture Note Series, No. 27. Cambridge University Press, Cambridge, 1977, MR 57:3190.
  • 5. Grothendieck, A.; Dieudonné, J. Éléments de Géométrie Algébrique I. Seconde édition, Springer-Verlag, Berlin, 1971, MR 36:177a.
  • 6. Hartshorne, Robin. Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, Berlin, 1977, MR 57:3116.
  • 7. Kaplansky, Irving. An introduction to differential algebra. Second edition. Actualités Scientifiques et Industrielles, No. 1251. Pulications de l'Institut de Mathématique de l'Université de Nancago, No. V. Hermann, Paris, 1996, MR 57:297.
  • 8. Keigher, William F. Adjunctions and comonads in differential algebra. Pacific J. Math. 59 (1975), no. 1, 99-112, MR 52:13770.
  • 9. - Prime differential ideals in differential rings. Contributions to algebra (collection of papers dedicated to Ellis Kolchin), 239-249. Academic Press, New York, 1977, MR 58:5610.
  • 10. Kolchin E. R. Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations. Ann. of Math. (2) 49, (1948). 1-42, MR 9:561c. (Reprinted in [14].)
  • 11. - Galois theory of differential fields. Amer. J. Math. 75, (1953). 753-824, MR 15:394a. (Reprinted in [14].)
  • 12. - Differential algebra and algebraic groups. Pure and Applied Mathematics, Vol. 54. Academic Press, New York-London, 1973, MR 58:27929
  • 13. - Constrained extensions of differential fields. Advances in Math. 12, (1974). 141-170, MR 49:4982. (Reprinted in [14].)
  • 14. Kolchin, Ellis. Selected works of Ellis Kolchin with commentary. Commentaries by Armand Borel, Michael F. Singer, Bruno Poizat, Alexandru Buium and Phyllis J. Cassidy. Edited and with a preface by Hyman Bass, Buium and Cassidy. American Mathematical Society, Providence, RI, 1999, MR 2000g:01042.
  • 15. Kolchin, Ellis; Lang, Serge. Algebraic groups and the Galois theory of differential fields. Amer. J. Math. 80 (1958). 103-110, MR 20:1109. (Reprinted in [14].)
  • 16. Kovacic, Jerald J. Differential schemes. In [39], pp. 71-94.
  • 17. - Global sections of diffspec, Journal of Pure and Applied Algebra, 171 (2002), No 2-3, 265-288, MR 2003c:12008.
  • 18. Levelt, A. H. M. Differential Galois theory and tensor products. Indag. Math. (N.S.) 1 (1990), no. 4, 439-449, MR 92f:12012.
  • 19. Mac Lane, Saunders. Categories for the working mathematician. Second edition. Graduate Texts in Mathematics, 5. Springer-Verlag, New York, 1998, MR 2001j:18001.
  • 20. Magid, Andy R. Lectures on differential Galois theory. University Lecture Series, 7. American Mathematical Society, Providence, RI, 1994, MR 95j:12008.
  • 21. Marker, David; Pillay, Anand. Differential Galois theory. III. Some inverse problems. Illinois J. Math. 41 (1997), no. 3, 453-461, MR 99m:12011.
  • 22. Mumford, David. Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay, Oxford University Press, London 1970, MR 44:219.
  • 23. Okugawa, Kôtaro. Differential algebra of nonzero characteristic. Lectures in Mathematics, 16. Kinokuniya Company Ltd., Tokyo, 1987, MR 92e:12007.
  • 24. Pillay, Anand. Differential Galois theory. I. Illinois J. Math. 42 (1998), no. 4, 678-699, MR 99m:12009.
  • 25. - Differential Galois theory. II. Joint AILA-KGS Model Theory Meeting (Florence, 1995). Ann. Pure Appl. Logic 88 (1997), no. 2-3, 181-191, MR 99m:12010.
  • 26. Poizat, Bruno. Une théorie de Galois imaginaire. J. Symbolic Logic 48 (1983), no. 4, 1151-1170, MR 85e:03083.
  • 27. - Stable groups. Translated from the 1987 French original by Moses Gabriel Klein. Mathematical Surveys and Monographs, 87. American Mathematical Society, Providence, RI, 2001, MR 2002a:03067.
  • 28. Scanlon, Thomas. Model theory and differential algebra. In [39], pp. 125-150, MR 2003g:03062.
  • 29. Sharp, Rodney Y. The dimension of the tensor product of two field extensions. Bull. London Math. Soc. 9 (1977), no. 1, 42-48, MR 55:10435.
  • 30. Silverman, Joseph H. The arithmetic of elliptic curves. Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1986. MR 87g:11070
  • 31. Sweedler, Moss. The predual theorem to the Jacobson-Bourbaki theorem. Trans. Amer. Math. Soc. 213 (1975), 391-406, MR 52:8188.
  • 32. Takeuchi, Mitsuhiro. A Hopf algebraic approach to the Picard-Vessiot theory. J. Algebra 122 (1989), no. 2, 481-509, MR 90j:12016.
  • 33. Umemura, Hiroshi. Galois theory of algebraic and differential equations. Nagoya Math. J. 144 (1996), 1-58, MR 98c:12009.
  • 34. - Differential Galois theory of infinite dimension. Nagoya Math. J. 144 (1996), 59-135, MR 98c:12010.
  • 35. Vámos, P. On the minimal prime ideals of a tensor product of two fields. Math. Proc. Cambridge Philos. Soc. 84 (1978), no. 1, 25-35, MR 80j:12016.
  • 36. van der Put, Marius. Differential Galois theory, universal rings and universal groups. In [39], pp. 171-189.
  • 37. Weil, André. Foundations of algebraic geometry. American Mathematical Society, Providence, R.I. 1962, MR 26:2439.
  • 38. Zariski, Oscar; Samuel, Pierre. Commutative algebra, Vol. 1. With the cooperation of I. S. Cohen. Corrected reprinting of the 1958 edition. Graduate Texts in Mathematics, No. 28. Springer-Verlag, New York-Heidelberg-Berlin, 1975, MR 52:5641.
  • 39. Guo, Li et al. (editors), Differential algebra and related topics (Proc. Internat. Workshop, Newark, NJ, 2000), World Sci. Publishing, River Edge, NJ, 2002, MR 2003b:12001.

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Additional Information

Jerald J. Kovacic
Affiliation: Department of Mathematics, The City College of The City University of New York, New York, New York 10031
Email: jkovacic@member.ams.org

DOI: https://doi.org/10.1090/S0002-9947-03-03306-3
Received by editor(s): June 1, 2002
Published electronically: July 2, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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