  We show that the Taylor-Wiles method can be applied to the cohomology of a
Shimura variety $S$ of PEL type attached to a unitary similitude group $G$,
with coefficients in the coherent sheaf attached to an automorphic vector
bundle $\CF$ , when $S$ has a smooth model over a $p$-adic integer ring. This
generalizes the main results of the article \cite{H13}, which treated the case
when $S$ is compact. As in the previous article, the starting point is a
theorem of Lan and Suh that proves the vanishing of torsion in the cohomology
under certain conditions on the parameters of the bundle $\CF$ and the prime
$p$. Most of the additional difficulty in the non-compact case is related to
showing that the contributions of boundary cohomology are all of Eisenstein
type. We also need to show that the coverings giving rise to the diamond
operators can be extended to \'etale coverings of appropriate toroidal
compactifications.
  The result is applied to show that, when the Taylor-Wiles method applies, the
congruence ideal attached to a coherent cohomological realization of an
automorphic Galois representation is independent of the signatures of the
hermitian form to which $G$ is attached. We also show that the Gorenstein
hypothesis used to construct $p$-adic $L$-functions in \cite{EHLS} is valid
under rather general hypotheses.

Number Theory