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), 417-437

MSC (2000):
Primary 13A35, 13N10, 16S32, 16W20, 16W22

Published electronically:
August 27, 2010

MathSciNet review:
2719688

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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**, https://doi.org/10.1090/S0273-0979-1982-15032-7**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**, https://doi.org/10.1090/S0002-9947-08-04567-4**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**, https://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**, https://doi.org/10.1090/S0002-9939-09-09825-6**5.**V. V. Bavula, The implies the , ArXiv:math. RA/0512250.**6.**Alexei Belov-Kanel 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****7.**Jacques Dixmier,*Sur les algèbres de Weyl*, Bull. Soc. Math. France**96**(1968), 209–242 (French). MR**0242897****8.**Yoshifumi Tsuchimoto,*Endomorphisms of Weyl algebra and 𝑝-curvatures*, Osaka J. Math.**42**(2005), no. 2, 435–452. MR**2147727**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
13A35,
13N10,
16S32,
16W20,
16W22

Retrieve articles in all journals with MSC (2000): 13A35, 13N10, 16S32, 16W20, 16W22

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:
https://doi.org/10.1090/S0002-9947-2010-05099-8

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