On a Question of Arveson about Ranks of Hilbert modules

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It's well known that the functional Hilbert space over the unit ball in Bd ∈ Cᵈ, with kernel function K(z,w)=11-z₁w₁-... -zdwd, admits a natural A(Bd)-module structure. We show the rank of a nonzero submodule is infinity if and only if the submodule is of infinite codimension. Together with Arveson's dilation theory, our result shows that Hilbert modules stand in stark contrast with Hilbert basis theorem for algebraic modules. This result answers a question of Arveson.