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GENERALIZATION OF McCONNELL’S THEOREM IN PRIME CHARACTERISTIC
Keywords:
Differential operators, Weyl algebras, $k$-algebra of finite typeDOI:
https://doi.org/10.17654/0972555525002Abstract
Let $k$ be a field and $I$ be an ideal of the polynomial algebra $R_n$ in $n$ variables. If $A$ is a (commutative) $k$-algebra, then we denote by $\mathcal{D}(A)$ the algebra of differential operators on the algebra $A$.
McConnell and Robson showed that in characteristic zero, the algebra of differential operators on a commutative $k$-algebra of finite type is isomorphic to a quotient of a subalgebra of the algebra of differential operators on a polynomial algebra, i.e.,
$$
\mathcal{D}\left(\frac{R_n}{I}\right) \simeq \frac{\left\{u \in \mathcal{D}\left(R_n\right): u(I) \subseteq I\right\}}{I \mathcal{D}\left(R_n\right)} .
$$
In this paper, we generalize this result of McConnell and Robson in prime characteristic.
Received: August 28, 2024
Accepted: October 21, 2024
References
A. Grothendieck and J. Dieudonné, Eléments de géométrie algébrique IV (quatrième partie), Publications Mathematiques I.H.E.S. 32 (1967), 39-40.
A. Sama and K. M. Kouakou, Quelques opérations algébriques dans l’ensemble des opérateurs différentiels sur une k-algèbre de type fini, Annales Mathématiques Africaines 7 (2018), 49-63.
A. Sama, Soro K. Fousséni and K. M. Kouakou, Algebra of differential operators on Laurent polynomials rings, RAMRES Sciences des Structures et de la Matière 7(2) (2023), 132-143.
J. Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968), 209-242.
J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Pure and Applied Mathematics, John Wiley & Sons, Chichester, UK, 1987, pp. 590-593.
K. E. Smith, Differential operators and connections with tight closure, Commutative Algebra, W. Bruns, J. Herzog, M. Hochster and U. Vetter, eds., Vechtaer Universitltsschriften, Cloppenburg, 1994, pp. 1755179.
L. Makar-Limanov, Rings of differential operators on algebraic curves, J. London. Math. Soc. 21 (1989), 538-540.
R. C. Cannings and M. P. Holland, Right ideals of rings of differential operators finite-dimensional algebra, J. Algebra 167 (1994), 116-141.
S. C. Countiho, A prime of algebraic D-modules, London Mathematical Society Student Texts 33, Cambridge University Press, 1995, pp. 22-23.
S. P. Smith and J. T. Stafford, Differential operators on an affine curve, Proc. London Math. Soc. 56 (1988), 229-259.
V. V. Bavula, Dimension, multiplicity, holonomic modules, and an analogue of the inequality of Bernstein for rings of differential operators in prime characteristic, Represent. Theory 13 (2009), 182-227. (Arxiv:math. RA/0605073).
William Nathaniel Traves, Differential operator and Nakai’s conjecture, Thèse Pour le diplôme de Doctorat en Philosophie, Département de Mathéatiques Université de Toronto, 1998, p. 20.
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