A question of Rentschler and the Dixmier problemA question of Rentschler and the Dixmier problem
Faculty of Sciences. Mathematics and Computer Science

Department of Mathematics - Computer Sciences

article

2001Princeton, N.J., 2001

Mathematics

Annals of mathematics. - Princeton, N.J.

154(2001):3, p. 683-702

0003-486X

000174207600004

E

English (eng)

University of Antwerp

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).

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