  When the quantum parameter $q^{1/2}$ is a root of unity of odd order. The
stated skein module $S_{q^{1/2}}(M,\mathcal{N})$ has an
$S_{1}(M,\mathcal{N})$-module structure, where $(M,\mathcal{N})$ is a marked
three manifold. We prove $S_{q^{1/2}}(M,\mathcal{N})$ is a finitely generated
$S_{1}(M,\mathcal{N})$-module when $M$ is compact, which furthermore indicates
the reduced stated skein module for the compact marked three manifold is finite
dimensional. We also give an upper bound for the dimension of
$S_{q^{1/2}}(M,\mathcal{N})$ over $S_{1}(M,\mathcal{N})$ when $M$ is compact.
  For a pb surface $\Sigma$, we use $S_{q^{1/2}}(\Sigma)^{(N)}$ to denote the
image of the Frobenius map when $q^{1/2}$ is a root of unity of odd order $N$.
Then $S_{q^{1/2}}(\Sigma)^{(N)}$ lives in the center of the stated skein
algebra $S_{q^{1/2}}(\Sigma)$. Let $\widetilde{S_{q^{1/2}}(\Sigma)^{(N)}}$ be
the field of fractions of $S_{q^{1/2}}(\Sigma)^{(N)}$, and
$\widetilde{S_{q^{1/2}}(\Sigma)}$ be
$S_{q^{1/2}}(\Sigma)\otimes_{S_{q^{1/2}}(\Sigma)^{(N)}}
\widetilde{S_{q^{1/2}}(\Sigma)^{(N)}}$. Then we show the dimension of
$\widetilde{S_{q^{1/2}}(\Sigma)}$ over $\widetilde{S_{q^{1/2}}(\Sigma)^{(N)}}$
is $N^{3r(\Sigma)}$ where $r(\Sigma)$ equals to the number of boundary
components of $\Sigma$ minus the Euler characteristic of $\Sigma$.
