Extensions of the Frobenius to the ring of differential operators on a polynomial algebra in prime characteristic
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- by V. V. Bavula
- Trans. Amer. Math. Soc. 363 (2011), 417-437
- DOI: https://doi.org/10.1090/S0002-9947-2010-05099-8
- Published electronically: August 27, 2010
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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).References
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Bibliographic Information
- V. V. Bavula
- Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
- MR Author ID: 293812
- Email: v.bavula@sheffield.ac.uk
- Received by editor(s): August 21, 2008
- Received by editor(s) in revised form: May 3, 2009
- Published electronically: August 27, 2010
- © Copyright 2010 American Mathematical Society
- 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
- MathSciNet review: 2719688