Spectra of modules over an affine scheme

In the present talk we intend to explain how spectra from the commutative algebra can be converted to the spectra of operator tuples. We demonstrate why the associated primes to a module can be treated as the joint eigenvalues, whereas support primes play the role of spectral values from Taylor spectrum. The gap between support and Taylor spectrum is described in terms of the scalar tuples over the ground field. In analysis that is the gap between Putinar and Taylor spectra, which is really complicated to explain. The algebraic spectral mapping theorem plays a central role in this theory. All the main key notions and properties from the commutative algebra will be provided.