Projective modules on Fock spaces

A Hilbert module over the free algebra generated by n noncommutative variables is a Hilbert space R with n bounded linear operators. In this paper we use Hilbert module language to study the semi-invariant subspaces of a family of weighted Fock spaces and their quotients that includes the Full Fock space, the symmetric Fock space, the Dirichlet algebra, and the reproducing kernel Hilbert spaces with a Nevanlinna-Pick kernel. We prove a commutant lifting theorem, obtain explicit resolutions and characterize the strongly orthogonally projective subquotients of each algebra. We use the symbols associated with the commutant lifting theorem to prove that two minimal projective resolutions are unitarily equivalent.