ABSTRACT: One approach to multivariate operator theory is via the study of Hilbert modules, that is , modules that have a Hilbert space structure over an algebra of holomorphic functions. Natural examples of Hilbert modules, which are already interesting, are obtained from ideals in the algebra of polynomials and the corresponding quotient modules.

We discuss some examples showing how the description of them involves ideas from algebraic and complex geometry. For the submodules arising from polynomial ideals a rigidity phenomenom can be demonstrated using a notion of localization while the quotient modules seem to be best described using concepts from complex geometry. We will focus on some of the simplest concrete examples to illustrate what is possible, attempting to keep prerequisites to a minimum.