Noncommutative localizations and joint spectra of a contractive quantum plane

In the present talk we discuss the localizations in the sense of J. L. Taylor (1972) of the Arens-Michael-Fréchet algebras occurred within the formal geometry of a contractive q-plane. It turns out that all noncommutative Fréchet algebras obtained by the Fréchet sheaves of the formal geometry are indeed localizations. That topological homology property of the structure sheaf results in the key properties of Taylor and Putinar spectra of the Banach q-modules. The main result is illustrated on a concrete example of a left Banach q-module to see a big gap between these spectra that makes the q-theory more elegant or even exotic.