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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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Abstract: Let $ K$ be a field of characteristic $ p>0$. It is proved that each automorphism $ \sigma \in \operatorname{Aut}_K(\mathcal D(P_n))$ of the ring $ \mathcal D(P_n)$ of differential operators on a polynomial algebra $ P_n= K[x_1, \ldots, x_n]$ is uniquely determined by the elements $ \sigma (x_1), \ldots ,\sigma (x_n)$, and that the set $ \operatorname{Frob}(\mathcal D(P_n))$ of all the extensions of the Frobenius (homomorphism) from certain maximal commutative polynomial subalgebras of $ \mathcal D(P_n)$, such as $ P_n$, to the ring $ \mathcal D(P_n)$ is equal to $ \operatorname{Aut}_K(\mathcal D(P_n) ) \cdot \mathcal{F}$ where $ \mathcal{F}$ is the set of all the extensions of the Frobenius from $ P_n$ to $ \mathcal D(P_n)$ that leave invariant the subalgebra of scalar differential operators. The set $ \mathcal{F}$ 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: http://dx.doi.org/10.1090/S0002-9947-2010-05099-8
PII: S 0002-9947(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