Abstract
It is usually admitted, even by the specialists of the Jacobian Conjecture, that it has no hope to be correctly formulated over fields of positive characteristic. This opinion is based on the well known counter-example F = X − X P of a polynomial in one indeterminate X over the prime field F p of cardinality p > 0, whose derivate is 1 and who does not define an automorphism of the F p -algebra F p [X]. But we could remark that the geometric degree of F, i.e. the dimension of the field F p (X) over F p (F), is a multiple of p. From our point of view, this fact is the only accident which could made the traditional formulation of the Jacobian Conjecture fall down in characteristic p. Hence, we think that it is sufficientce to avoid this accident to obtain the right and universal formulation of the classical Jacobian conjecture for the automorphisms of the algebras of polynomials in any number of polynomials over any domain of any characteristic (see its precise statement in 3.1).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Altman and S. Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, vol. 276, Springer-Verlag, New York, 1970.
M. Auslander and D.A. Buchsbaum, On ramification theory in Noetherian rings, Amer. J. Math. 81(1959), 749–765.
H. Bass, Differential Structure of Etale Extensions of Polynomial Algebras, Commutative Algebra (M. Hochster, C. Huneke, and J.D. Sally, eds.), Springer-Verlag, New York, 1989, Proceedings of a Microprogram Hold, June 15–July 2, 1987.
H. Bass, E. Connell, and D. Wright, The Jacobian Conjecture: Reduction of Degreeand Formal Expansion of the Inverse, Bulletin of the American Mathematical Society 7(1982), 287–330.
N. Bourbaki, Algebre, Hermann, Paris, ch. 4.
F. Demeyer and E. Ingraham, Separable Algebras over Commutative Rings, Lecture Notes in Mathematics, vol. 181, Springer-Verlag, Berlin-Heidelberg-New-York, 1971.
E. Formanek, Two notes on the Jacobian Conjecture, Arch. Math. 49 (1967), 286291.
A. Grothendieck, Seminaire de Güomütrie Algübrique, 1960–1961, I.H.E.S., 1960, ch. I, Fascicule 1, 3üme üdition corrigüe.
A. Grothendieck and J. Dieudonne, Elements de Geomütrie Algübrique IV, 64).
A. Grothendieck and J. Dieudonne, Elements de Geométrie Algébrique IV, Publ.Math. I.H.E.S. 20(1965).
A. Grothendieck and J Dieudonne, Elements de Geométrie Algébrique IV, Publ.Math. I.H.E.S. 28(1966).
A. Grothendieck and J Dieudonne, Elements de Geométrie Algébrique IV, Publ.Math. I.H.E.S. 32(1967).
M.-A. Knus and M. Ojanguren, Theorie dela Descente et Algèbresd’Azumaya, Lecture Notes in Mathematics, vol. 399, Springer-Verlag, Berlin, 1974.
M.-P. Malliavin, Algèbre Commutative, Masson, Paris, 1985.
H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1989, Cambridge Studies in Advanced Mathematics 8.
R. Morris and S.S.-S. Wang, A Jacobian Criterion for Smoothness, J. of Algebra 69(1981), 483–486.
M. Nagata, Flatness ofan extension of a commutativering, J. Math. Kyoto Univ. 9(1969), 439–448.
H. Seydi, Un théorème de descente effective universelle etune application, Serie A, C.R. Acad. Sc. Paris, 1970, t. 270 (ler avril 1970 ).
S.S.-S. Wang, A Jacobian criterion for separability, J. of Algebra 65 (1980), 453–494, MR 83e:14010, Zb1.471.13005.
D. Wright, On the Jacobian Conjecture, IllinoisJ. of Math. 15(1981), no. 3, 423–440.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Adjamagbo, K. (1995). On Separable Algebras over a U.F.D. and the Jacobian Conjecture in Any Characteristic. In: van den Essen, A. (eds) Automorphisms of Affine Spaces. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8555-2_5
Download citation
DOI: https://doi.org/10.1007/978-94-015-8555-2_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4566-9
Online ISBN: 978-94-015-8555-2
eBook Packages: Springer Book Archive