The finite dual coalgebra as a quantization of the maximal spectrum

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 • act as an appropriate depiction of the noncommutative maximal spectrum of A; importantly, this class includes affine noetherian PI algebras. We investigate cases where the finite dual coalgebra of a twisted tensor product is a crossed product coalgebra of the respective finite duals. This is achieved by interpreting the finite dual as a topological dual. Sufficient conditions for this result to be applied to Ore extensions, smash product algebras, and crossed product bialgebras are described. 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. Finally, we implement these techniques for quantum planes at roots of unity as an endeavor to visualize the noncommutative space on which these algebras act as functions.