Title
A question of Rentschler and the Dixmier problemA question of Rentschler and the Dixmier problem
Author
Faculty/Department
Faculty of Sciences. Mathematics and Computer Science
Research group
Department of Mathematics - Computer Sciences
Publication type
article
Publication
Princeton, N.J.,
Subject
Mathematics
Source (journal)
Annals of mathematics. - Princeton, N.J.
Volume/pages
154(2001):3, p. 683-702
ISSN
0003-486X
ISI
000174207600004
Carrier
E
Target language
English (eng)
Full text (Publishers DOI)
Affiliation
University of Antwerp
Abstract
THEOREM. Let R and T be somewhat commutative algebras with the same holonomic number and let phi : R --> T be an algebra homomorphism. Then every holonomic T-module M is (via phi) a holonomic R-module and has finite length as an R-module. When applied to the Weyl algebra this result gives a positive answer to a question of Rentschler. In the important case where R = D(X) and T = D(Y) are rings of differential operators on smooth irreducible algebraic affine varieties X and Y of the same dimension the result means that holonomicity is preserved by twisting (by an arbitrary algebra homomorphism). A short proof is given of the well-known fact that an affirmative solution to the Dixmier Problem (whether every algebra endomorphism phi of the Weyl algebra A,, is an automorphism) implies the Jacobian Conjecture. The Dixmier Problem has a positive answer if and only if the (twisted from both sides) A(n)-bimodule (phi)A(n)(phi) is simple for each phi. (To begin with, it is not even clear whether it is finitely generated). The theorem implies that it has finite length (moreover, the bimodule is holonomic, thus simple subfactors of it have the least possible Gelfand-Kirillov dimension).
E-info
https://repository.uantwerpen.be/docman/iruaauth/95e23f/5ba2004.pdf
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