Deformation quantization and noncommutative formal geometry of a contractive quantum plane

We discuss the noncommutative formal geometry of a contractive quantum plane, whose spectrum is the union of two copies of the complex plane. It turns out that a formal completion of the Arens-Michael envelope of a contractive quantum plane provides the noncommutative analytic space, whose base topological space is the same spectrum, whereas the structure sheaf is obtained as the deformation quantization of the related commutative analytic space. As the basic tool we use the fibered products of the Fréchet sheaves. The related topological homology problems are considered to find out a key link between the transversality relation of the noncommutative sections versus to a left Fréchet module, and noncommutative Taylor spectrum of the module.