Abstract
By a theorem of Chevalley the image of a morphism of varieties is a constructible set. The algebraic version of this fact is usually stated as a result on “extension of specializations” or “lifting of prime ideals”. We present a difference analog of this theorem. The approach is based on the philosophy that occasionally one needs to pass to higher powers of σ, where σ is the endomorphism defining the difference structure. In other words, we consider difference pseudo fields (which are finite direct products of fields) rather than difference fields. We also prove a result on compatibility of pseudo fields and present some applications of the main theorem, e.g. constrained extension and uniqueness of differential Picard–Vessiot rings with a difference parameter.
Similar content being viewed by others
References
Amano K., Masuoka A.: Picard-Vessiot extensions of Artinian simple module algebras. J. Algebra 285(2), 743–767 (2005). doi:10.1016/j.jalgebra.2004.12.006
Antieau, B., Ovchinnikov, A., Trushin, D.: Galois theory of difference equations with periodic parameters (2010). ArXiv:1009.1159v1
Atiyah M.F., Macdonald I.G.: Introduction to commutative algebra. Addison-Wesley, Reading (1969)
Bourbaki, N.: Algebra II. Chapters 4–7. Elements of Mathematics (Berlin). Springer, Berlin (1990) (Translated from the French by P.M. Cohn and J. Howie)
Cassidy, P.J., Singer, M.F.: Galois theory of parameterized differential equations and linear differential algebraic groups. In: Differential Equations and Quantum Groups. IRMA Lect. Math. Theor. Phys., vol. 9, pp. 113–155. Eur. Math. Soc., Zürich (2007)
Chatzidakis Z., Hrushovski E.: Model theory of difference fields. Trans. Am. Math. Soc. 351(8), 2997–3071 (1999). doi:10.1090/S0002-9947-99-02498-8
Chatzidakis, Z., Hrushovski, E., Peterzil, Y.: Model theory of difference fields. II. Periodic ideals and the trichotomy in all characteristics. Proc. Lond. Math. Soc. (3) 85(2), 257–311 (2002). doi:10.1112/S0024611502013576
Cohn R.M.: Difference Algebra. Interscience Publishers/Wiley, New York (1965)
Giral J.M.: Krull dimension, transcendence degree and subalgebras of finitely generated algebras. Arch. Math. (Basel) 36(4), 305–312 (1981). doi:10.1007/BF01223706
Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I. Inst. Hautes Études Sci. Publ. Math. 20, 259 (1964)
Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III. Inst. Hautes Études Sci. Publ. Math. 28, 255 (1966)
Hardouin C., Singer M.F.: Differential Galois theory of linear difference equations. Math. Ann. 342(2), 333–377 (2008). doi:10.1007/s00208-008-0238-z
Hardouin, C., di Vizio, L., Michael, W.: Difference Galois theory for linear differential equations. In preparation
Hrushovski, E.: The elementary theory of the Frobenius automorphisms. ArXiv:math/0406514v1. Updated version from http://www.ma.huji.ac.il/~ehud/ (2004)
Kac V.G.: A differential analog of a theorem of Chevalley. Int. Math. Res. Notices 13, 703–710 (2001). doi:10.1155/S1073792801000368
Kolchin, E.R.: Differential Algebra and Algebraic Groups. Academic Press, New York (1973). Pure and Applied Mathematics, vol. 54
Kolchin E.R.: Constrained extensions of differential fields. Adv. Math. 12, 141–170 (1974)
Kovacic J.J.: The differential Galois theory of strongly normal extensions. Trans. Am. Math. Soc. 355(11), 4475–4522 (2003). doi:10.1090/S0002-9947-03-03306-3
Landesman P.: Generalized differential Galois theory. Trans. Am. Math. Soc. 360(8), 4441–4495 (2008). doi:10.1090/S0002-9947-08-04586-8
Levin, A.: Difference algebra. In: Algebra and Applications, vol. 8. Springer, New York (2008)
Marker, D., Messmer, M., Pillay, A.: Model theory of fields. In: Lecture Notes in Logic, vol. 5, 2nd edn. Association for Symbolic Logic, La Jolla (2006)
Matzat, B.H.: Differential Galois theory in positive characteristic. IWR-Preprint (2001)
Okugawa, K.: Differential algebra of nonzero characteristic. In: Lectures in Mathematics, vol. 16. Kinokuniya Company Ltd., Tokyo (1987)
van der Put, M., Singer, M.F.: Galois theory of difference equations. In: Lecture Notes in Mathematics, vol. 1666. Springer, Berlin (1997)
Rosen, E.: A differential Chevalley theorem. ArXiv:0810.5486
Trushin, D.: Difference Nullstellensatz. ArXiv:0908.3865v1 (2009)
Trushin, D.: Difference Nullstellensatz in the case of finite group. ArXiv:0908.3863v1 (2009)
Wibmer, M.: Geometric difference Galois theory. Ph.D. thesis, Heidelberg (2010). http://www.ub.uni-heidelberg.de/archiv/10685