Publication
Title
A question of Rentschler and the Dixmier problem
Author
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).
Language
English
Source (journal)
Annals of mathematics. - Princeton, N.J.
Publication
Princeton, N.J. : 2001
ISSN
0003-486X
Volume/pages
154:3(2001), p. 683-702
ISI
000174207600004
Full text (Publishers DOI)
Full text (publishers version - intranet only)
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