  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.
