The Dixmier-Moeglin equivalence, Morita equivalence, and homeomorphism of spectra

Abstract

Let $k$ be a field and let $R$ be a left noetherian $k$-algebra. The algebra $R$ satisfies the Dixmier-Moeglin equivalence if the annihilators of irreducible representations are precisely those prime ideals that are locally closed in the ${\rm Spec}(R)$ and if, moreover, these prime ideals are precisely those whose extended centres are algebraic extensions of the base field. We show that if $R$ and $S$ are two left noetherian $k$-algebras with ${\rm dim}_k(R), {\rm dim}_k(S)<|k|$ then if $R$ and $S$ have homeomorphic spectra then $R$ satisfies the Dixmier-Moeglin equivalence if and only if $S$ does. In particular, the topology of ${\rm Spec}(R)$ can detect the Dixmier-Moeglin equivalence in this case. In addition, we show that if $k$ is uncountable and $R$ is affine noetherian and its prime spectrum is a disjoint union of subspaces that are each homeomorphic to the spectrum of an affine commutative ring then $R$ satisfies the Dixmier-Moeglin equivalence. We show that neither of these results need hold if $k$ is countable and $R$ is infinite-dimensional. Finally, we make the remark that satisfying the Dixmier-Moeglin equivalence is a Morita invariant and finally we show that $R$ and $S$ are left noetherian $k$-algebras that satisfy the Dixmier-Moeglin equivalence then $R\otimes_k S$ does too, provided it is left noetherian and satisfies the Nullstellensatz; and we show that $eRe$ also satisfies the Dixmier-Moeglin equivalence, where $e$ is a nonzero idempotent of $R$.