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 Tmodule M is (via phi) a holonomic Rmodule and has finite length as an Rmodule. 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 wellknown 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 GelfandKirillov dimension).   
Language 



English
 
Source (journal) 



Annals of mathematics.  Princeton, N.J.  
Publication 



Princeton, N.J. : 2001
 
ISSN 



0003486X
 
Volume/pages 



154:3(2001), p. 683702
 
ISI 



000174207600004
 
Full text (Publisher's DOI) 


  
Full text (publisher's version  intranet only) 


  
