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Isomorphisms of Rings of Differential Operators on Curves

Isomorphisms of Rings of Differential Operators on Curves ISOMORPHISMS OF RINGS OF DIFFERENTIAL OPERATORS ON CURVES PATRICK PERKINS 0. Introduction Let X and Y be nonisomorphic irreducible affine algebraic curves over the complex numbers C. Let D{X) and D{Y) be their rings of differential operators (see [7] for the definition). The works of Makar-Limanov [3] and the author [5] address the question of when D(X) is isomorphic to D(Y). It is shown that if D(X) s D(Y) then O(X) and 0(7) have integral closure C[/], and that the normalization map n: X -* X is injective. In [2] Makar-Limanov and G. Letzter show further that D(X) and D(Y) must have the same codimension in the graded ring associated with the order of differential operator filtration. This allows them to distinguish between many rings of differential operators. In this paper we give a criterion for two rings of differential operators on curves to be isomorphic, and apply it to a large family of curves. The results extend those of G. Letzter in [1], but the methods are quite different. We say X is a monomial curve if O(X) £ C[t] is spanned by the monomials it contains. If .Af is a monomial curve then D(X) can be considered http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Isomorphisms of Rings of Differential Operators on Curves

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/23.2.133
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Abstract

ISOMORPHISMS OF RINGS OF DIFFERENTIAL OPERATORS ON CURVES PATRICK PERKINS 0. Introduction Let X and Y be nonisomorphic irreducible affine algebraic curves over the complex numbers C. Let D{X) and D{Y) be their rings of differential operators (see [7] for the definition). The works of Makar-Limanov [3] and the author [5] address the question of when D(X) is isomorphic to D(Y). It is shown that if D(X) s D(Y) then O(X) and 0(7) have integral closure C[/], and that the normalization map n: X -* X is injective. In [2] Makar-Limanov and G. Letzter show further that D(X) and D(Y) must have the same codimension in the graded ring associated with the order of differential operator filtration. This allows them to distinguish between many rings of differential operators. In this paper we give a criterion for two rings of differential operators on curves to be isomorphic, and apply it to a large family of curves. The results extend those of G. Letzter in [1], but the methods are quite different. We say X is a monomial curve if O(X) £ C[t] is spanned by the monomials it contains. If .Af is a monomial curve then D(X) can be considered

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Mar 1, 1991

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