  In this article, we use deformation theory of Galois representations valued
in the symplectic group of degree four to prove a freeness result for the
cohomology of certain quaternionic unitary Shimura variety over the universal
deformation ring for certain type of residual representation satisfying a
property called rigidity. This result plays an important role in the proof of
the arithmetic level raising theorem for the symplectic similitude group of
degree four over the field of rational numbers by the author.
