A note on skew differential operators on commutative rings

Let A be a commutative integral domain that is a finitely generated algebra over a field k of characteristic 0 and let ø be a k-algebra automorphism of A of finite order m. In this note we study the ring D(A;ø of differential operators introduced by A.D. Bell. We prove that if A is a free module ove...

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Bibliographic Details
Published in:Communications in algebra 2000-01, Vol.28 (8), p.3777-3784
Main Authors: Hirano, Yasuyuki, NasuKentaro Tsuda, Kouji, Tsuda, Kentaro
Format: Article
Language:English
Subjects:
Primary: 16S32
Secondary: 16S36
Online Access:Get full text
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Summary:Let A be a commutative integral domain that is a finitely generated algebra over a field k of characteristic 0 and let ø be a k-algebra automorphism of A of finite order m. In this note we study the ring D(A;ø of differential operators introduced by A.D. Bell. We prove that if A is a free module over the fixed sub-ring A ø , with a basis containing 1, then D(A;ø) is isomorphic to the matrix ring M m (D(A ø ). It follows from Grothendieck's Generic Flatness Theorem that for an arbitrary A there is an element cϵAsuch that D(A[c -1 ];ø)≅M m (D(A[c -1 ] ø )). As an application, we consider the structure of D(A;ø)when A is a polynomial or Laurent polynomial ring over k and ø is a diagonalizable linear automorphism.
ISSN:0092-7872
1532-4125
DOI:10.1080/00927870008827056