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

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

Published electronically:
August 27, 2010

MathSciNet review:
2719688

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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).

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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:
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