  In this paper, we study the Kauffman bracket skein module of closed oriented
three-manifolds at a non-multiple-of-four roots of unity. Our main result
establishes that the localization of this module at a maximal ideal, which
corresponds to an irreducible representation of the fundamental group of the
manifold, forms a one-dimensional free module over the localized unreduced
coordinate ring of the character variety. We apply this by proving that the
dimension of the skein module of a homology sphere with finite character
variety, when the order of the root of unity is not divisible by $4$, is
greater than or equal to the dimension of the unreduced coordinate ring of the
character variety. This leads to a computation of the dimension of the skein
module with coefficients in rational functions for homology spheres with tame
universal skein module.
