  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.
