The fredholm index of quotient Hilbert modules

We show that the (multivariable) Fredholm index of a broad class of quotient Hilbert modules can be calculated by the Samuel multiplicity. These quotient modules include the Hardy, Bergman, or symmetric Fock spaces in several variables modulo submodules generated by multipliers. Our calculation is based on Gleason, Richter, Sundberg's results on the Fredholm index of the corresponding submodules. However, our main result (Theorem 2) establishes a formula with independent interests in a broader context. It relates the fibre dimension of a submodule, an analytic notion, to the Samuel multiplicity of the quotients module, an algebraic notion. When applied to multivariable Fredholm theory, we establish the following general principle which yields the above calculation of indices as a special case: The Fredholm index of a submodule is equal to its fibre dimension if and only if the index of the quotient module is equal to its Samuel multiplicity.