Two properties of endomorphisms of Weyl algebras

TL;DR

This paper investigates the properties of endomorphisms of Weyl algebras, proving flatness and automorphism conditions in characteristic zero, and providing counterexamples in positive characteristic.

Contribution

It establishes flatness and automorphism criteria for Weyl algebra endomorphisms over characteristic zero fields, and demonstrates limitations in positive characteristic.

Findings

Endomorphisms of Weyl algebras over characteristic zero are flat.

Birational endomorphisms are automorphisms in characteristic zero.

Counterexamples show non-flatness in positive characteristic.

Abstract

We show that endomorphisms of Weyl algebras over fields of characteristic zero are flat and that birational endomorphisms are automorphisms by reducing to positive characteristic. We also give examples showing that endomorphisms of Weyl algebras are not in general flat over fields of positive characteristic.