A Question of Rentschler and the Dixmier Problem

Abstract

The following theorem is the main result of the paper. THEOREM. Let R and T be somewhat commutative algebras with the same holonomic number and let φ: R → T be an algebra homomorphism. Then every holonomic T-module M is (via φ) 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 φ of the Weyl algebra An 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 φAnφ is simple for each φ. (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).