The finite dual coalgebra as a quantization of the maximal spectrum

TL;DR

This paper explores how the Heyneman-Sweedler finite dual coalgebra can serve as a quantized version of the maximal spectrum in noncommutative geometry, especially for certain classes of algebras like affine noetherian PI algebras.

Contribution

It introduces fully residually finite-dimensional algebras as a framework for representing noncommutative spectra via finite duals, extending the understanding of quantum planes and maximal orders.

Findings

Finite dual coalgebra models noncommutative spectra.

Description of Azumaya locus in finite duals of prime affine algebras.

Application to quantum planes at roots of unity.

Abstract

In pursuit of a noncommutative spectrum functor, we argue that the Heyneman-Sweedler finite dual coalgebra can be viewed as a quantization of the maximal spectrum of a commutative affine algebra, integrating prior perspectives of Takeuchi, Batchelor, Kontsevich-Soibelman, and Le Bruyn. We introduce fully residually finite-dimensional algebras $A$ as those with enough finite-dimensional representations to let $A^\circ$ act as an appropriate depiction of the noncommutative maximal spectrum of $A$; importantly, this class includes affine noetherian PI algebras. In the case of prime affine algebras that are module-finite over their center, we describe how the Azumaya locus is represented in the finite dual. This is used to describe the finite dual of quantum planes at roots of unity as an endeavor to visualize the noncommutative space on which these algebras act as functions. Finally, we discuss how a similar analysis can be carried out for other maximal orders over surfaces.