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Finitely generated bimodules over Weyl algebras

Niels Lauritzen, Jesper Funch Thomsen

Abstract

Let $A$ be the $n$-th Weyl algebra over a field of characteristic zero, and $\varphi:A\rightarrow A$ an endomorphism with $S = \varphi(A)$. We prove that if $A$ is finitely generated as a left or right $S$-module, then $S = A$. The proof involves reduction to large positive characteristics. By holonomicity, $A$ is always finitely generated as an $S$-bimodule. Moreover, if this bimodule property could be transferred into a similar property in large positive characteristics, then we could again conclude that $A=S$. The latter would imply the Dixmier Conjecture.

Finitely generated bimodules over Weyl algebras

Abstract

Let be the -th Weyl algebra over a field of characteristic zero, and an endomorphism with . We prove that if is finitely generated as a left or right -module, then . The proof involves reduction to large positive characteristics. By holonomicity, is always finitely generated as an -bimodule. Moreover, if this bimodule property could be transferred into a similar property in large positive characteristics, then we could again conclude that . The latter would imply the Dixmier Conjecture.
Paper Structure (11 sections, 18 theorems, 48 equations)

This paper contains 11 sections, 18 theorems, 48 equations.

Table of Contents

  1. Ultraproducts
  2. Filters
  3. Ultraproducts
  4. The Weyl algebra
  5. The Weyl algebra in positive characteristic
  6. Endomorphisms of Weyl algebras
  7. The main result
  8. Bounds on polynomial endomorphisms
  9. The classical bound on the degree of the inverse
  10. Integral extensions
  11. A Gröbner basis approach

Key Result

Proposition 1.1

Let $R$ be an integral domain with $\operatorname{Jac}(R) = \{0\}$, $X = \operatorname{Specm}(R)$ and $X_r = \{\mathfrak{m}\in X\mid r\notin \mathfrak{m}\}$ for $r\in R$. Then is a filter on $X$. Let $\mathcal{U}$ be an ultrafilter on $X$ containing $\mathcal{F}$. Then $\prod_{\mathcal{U}} R/\mathfrak{m}$ contains the fraction field of $R$.

Theorems & Definitions (38)

  • Proposition 1.1
  • proof
  • Lemma 1.2
  • proof
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • ...and 28 more