Characteristic Spaces and Rigidity for Analytic Hilbert Modules

It is well known that a polynomial in one variable is completely determined by its zeros (counting multiplicities). We generalize this result to an ideal of polynomials in several variables by introducing the characteristic spaces of the ideal. One finds that the ideal is completely determined by its characteristic spaces on a characteristic set. In particular, a primary ideal is completely determined by its characteristic space at any zero point. Some straightforward applications of the above results yield the algebraic reduction theorem for analytic Hilbert modules in several variables. Also, we obtain some general rigidity results for analytic Hilbert modules by using the techniques of AF-envelopes of analytic Hilbert modules.