On the Unilateral Shift as a Hilbert Module over the Disc Algebra

We study the unilateral shift (of arbitrary countable multiplicity) as a Hilbert module over the disc algebra and the associated extension groups. In relation with the problem of determining whether this module is projective, we consider a special class of extensions, which we call polynomial. We show that the subgroup of polynomial extensions of a contractive module by the adjoint of the unilateral shift is trivial. The main tool is a function theoretic decomposition of the Sz.-Nagy-Foias model space for completely non-unitary contractions.