Noncommutative formal geometry of a contractive quantum plane

The present talk is devoted to the noncommutative complex analytic geometry of a contractive quantum plane from the formal completion point of view. The formal completion of an Arens-Michael envelope of the quantum plane possesses the same spectrum to be the union of two copies of the complex plane. It turns out that it can be extended up to an Arens-Michael-Fréchet algebra sheaf, which results in the noncommutative analytic space, whose base topological space is the same spectrum. Moreover, that sheaf can be obtained as the deformation quantization of the related commutative analytic space. As the basic tool we use the fibered products of the Fréchet sheaves. To find out a key link between the transversality relation of the noncommutative sections versus to a left Fréchet module we discuss the related topological homology problems, and the related noncommutative Taylor spectrum of the module.

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