The Dixmier-Moeglin equivalence for extensions of scalars and Ore extensions

TL;DR

This paper investigates the stability of the Dixmier-Moeglin equivalence property in noncommutative algebra under scalar extensions and Ore extensions, establishing conditions under which this property is preserved.

Contribution

It proves that the Dixmier-Moeglin equivalence is maintained under base change for certain algebras and characterizes when Ore extensions preserve this equivalence, especially for frame-preserving automorphisms and derivations.

Findings

Dixmier-Moeglin equivalence is preserved under base change for finitely generated complex noetherian algebras.

Ore extensions with frame-preserving automorphisms or derivations maintain the Dixmier-Moeglin equivalence under specific conditions.

The property holds for Ore extensions of algebras with finite Gelfand-Kirillov dimension and all prime ideals being completely prime.

Abstract

An algebra $A$ satisfies the Dixmier-Moeglin equivalence if we have the equivalences: $$P~{\rm primitive}\iff P~{\rm rational}\iff P ~{\rm locally~closed~}\qquad~{\rm for}~P\in {\rm Spec}(A).$$ We study the robustness of the Dixmier-Moeglin equivalence under extension of scalars and under the formation of Ore extensions. In particular, we show that the Dixmier-Moeglin equivalence is preserved under base change for finitely generated complex noetherian algebras. We also study Ore extensions of finitely generated complex noetherian algebras $A$. If $T:A\to A$ is either a $\mathbb{C}$-algebra automorphism or a $\mathbb{C}$-linear derivation of $A$, we say that $T$ is \emph{frame-preserving} if there exists a finite-dimensional subspace $V\subseteq A$ that generates $A$ as an algebra such that $T(V)\subseteq V$. We show that if $A$ is of finite Gelfand-Kirillov dimension and has the property that all prime ideals of $A$ are completely prime and $A$ satisfies the Dixmier-Moeglin equivalence then the Ore extension $A[x;T]$ satisfies the Dixmier-Moeglin equivalence whenever $T$ is a frame-preserving derivation or automorphism.