The Dixmier-Moeglin equivalence for twisted homogeneous coordinate rings

TL;DR

This paper investigates the primitive spectrum of twisted homogeneous coordinate rings associated with projective schemes, establishing the Dixmier-Moeglin equivalence for dimensions up to two over algebraically closed fields.

Contribution

It proves the Dixmier-Moeglin equivalence for primitive ideals of twisted homogeneous coordinate rings when the scheme's dimension is at most two.

Findings

Primitive ideals are characterized by Dixmier-Moeglin conditions in the specified setting.

The result applies to twisted homogeneous coordinate rings over algebraically closed fields of characteristic zero.

The study extends understanding of noncommutative projective algebraic geometry in low dimensions.

Abstract

Given a projective scheme $X$ over a field $k$, an automorphism $\sigma$ of $X$, and a $\sigma$-ample invertible sheaf $L$, one may form the twisted homogeneous coordinate ring $B = B(X, L, \sigma)$, one of the most fundamental constructions in noncommutative projective algebraic geometry. We study the primitive spectrum of $B$, as well as that of other closely related algebras such as skew and skew-Laurent extensions of commutative algebras. Over an algebraically closed, uncountable field $k$ of characteristic zero, we prove that that the primitive ideals of $B$ are characterized by the usual Dixmier-Moeglin conditions whenever the dimension of $X$ is no more than 2.