Extensions of the Frobenius to the ring of differential operators on a polynomial algebra in prime characteristic
Author:
V. V. Bavula
Journal:
Trans. Amer. Math. Soc. 363 (2011), 417437
MSC (2000):
Primary 13A35, 13N10, 16S32, 16W20, 16W22
Published electronically:
August 27, 2010
MathSciNet review:
2719688
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Abstract: Let be a field of characteristic . It is proved that each automorphism of the ring of differential operators on a polynomial algebra is uniquely determined by the elements , and that the set of all the extensions of the Frobenius (homomorphism) from certain maximal commutative polynomial subalgebras of , such as , to the ring is equal to where is the set of all the extensions of the Frobenius from to that leave invariant the subalgebra of scalar differential operators. The set is found explicitly; it is large (a typical extension depends on countably many independent parameters).
 1.
Hyman
Bass, Edwin
H. Connell, and David
Wright, The Jacobian conjecture: reduction of
degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 287–330. MR 663785
(83k:14028), http://dx.doi.org/10.1090/S027309791982150327
 2.
V.
V. Bavula, Simple derivations of differentiably
simple Noetherian commutative rings in prime characteristic, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4007–4027. MR 2395162
(2009c:13060), http://dx.doi.org/10.1090/S0002994708045674
 3.
V.
V. Bavula, The inversion formulae for automorphisms of polynomial
algebras and rings of differential operators in prime characteristic,
J. Pure Appl. Algebra 212 (2008), no. 10,
2320–2337. MR 2426512
(2009j:13030), http://dx.doi.org/10.1016/j.jpaa.2008.03.009
 4.
V.
V. Bavula, The group of order preserving
automorphisms of the ring of differential operators on a Laurent polynomial
algebra in prime characteristic, Proc. Amer.
Math. Soc. 137 (2009), no. 6, 1891–1898. MR 2480268
(2010h:16057), http://dx.doi.org/10.1090/S0002993909098256
 5.
V. V. Bavula, The implies the , ArXiv:math. RA/0512250.
 6.
Alexei
BelovKanel and Maxim
Kontsevich, The Jacobian conjecture is stably equivalent to the
Dixmier conjecture, Mosc. Math. J. 7 (2007),
no. 2, 209–218, 349 (English, with English and Russian
summaries). MR
2337879 (2009f:16041)
 7.
Jacques
Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math.
France 96 (1968), 209–242 (French). MR 0242897
(39 #4224)
 8.
Yoshifumi
Tsuchimoto, Endomorphisms of Weyl algebra and
𝑝curvatures, Osaka J. Math. 42 (2005),
no. 2, 435–452. MR 2147727
(2006g:14101)
 1.
 H. Bass, E. H. Connell and D. Wright, The Jacobian Conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (New Series), 7 (1982), 287330. MR 663785 (83k:14028)
 2.
 V. V. Bavula, Simple derivations of differentiably simple Noetherian commutative rings in prime characteristic, Trans. Amer. Math. Soc., 360 (2008), no. 8, 40074027. MR 2395162 (2009c:13060)
 3.
 V. V. Bavula, The inversion formulae for automorphisms of polynomial algebras and differential operators in prime characteristic, J. Pure Appl. Algebra, 212 (2008), 23202337. MR 2426512
 4.
 V. V. Bavula, The group of order preserving automorphisms of the ring of differential operators on a Laurent polynomial algebra in prime characteristic, Proc. Amer. Math. Soc., 137 (2009), 18911898. MR 2480268
 5.
 V. V. Bavula, The implies the , ArXiv:math. RA/0512250.
 6.
 A. BelovKanel and M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier Conjecture, Mosc. Math. J., 7 (2007), no. 2, 209218. MR 2337879 (2009f:16041)
 7.
 J. Dixmier, Sur les algèbres de Weyl. Bull. Soc. Math. France, 96 (1968), 209242. MR 0242897 (39:4224)
 8.
 Y. Tsuchimoto, Endomorphisms of Weyl algebra and curvatures. Osaka J. Math., 42 (2005), no. 2, 435452. MR 2147727 (2006g:14101)
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Additional Information
V. V. Bavula
Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
Email:
v.bavula@sheffield.ac.uk
DOI:
http://dx.doi.org/10.1090/S000299472010050998
PII:
S 00029947(2010)050998
Keywords:
Extensions of the Frobenius,
ring of differential operators,
Frobenius polynomial subalgebra,
group of automorphisms
Received by editor(s):
August 21, 2008
Received by editor(s) in revised form:
May 3, 2009
Published electronically:
August 27, 2010
Article copyright:
© Copyright 2010 American Mathematical Society
