Finiteness and dimension of stated skein modules over Frobenius

When the quantum parameter $q^{1\mathrm{/}2}$ is a root of unity of odd order. The stated skein module $S_{q^{1\mathrm{/}2}}(M,\mathcal{N})$ has an $S_1(M,\mathcal{N})$-module structure, where $(M,\mathcal{N})$ is a marked three manifold. We prove $S_{q^{1\mathrm{/}2}}(M,\mathcal{N})$ 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 $S_{q^{1\mathrm{/}2}}(M,\mathcal{N})$ over $S_1(M,\mathcal{N})$ when $M$ is compact. For a pb surface $\mathrm{\Sigma }$, we use $S_{q^{1\mathrm{/}2}}(\mathrm{\Sigma }{)}^{(N)}$ to denote the image of the Frobenius map when $q^{1\mathrm{/}2}$ is a root of unity of odd order $N$. Then $S_{q^{1\mathrm{/}2}}(\mathrm{\Sigma }{)}^{(N)}$ lives in the center of the stated skein algebra $S_{q^{1\mathrm{/}2}}(\mathrm{\Sigma })$. Let $\stackrel{~}{S_{q^{1\mathrm{/}2}}(\mathrm{\Sigma }{)}^{(N)}}$ be the field of fractions of $S_{q^{1\mathrm{/}2}}(\mathrm{\Sigma }{)}^{(N)}$, and $\stackrel{~}{S_{q^{1\mathrm{/}2}}(\mathrm{\Sigma })}$ be $S_{q^{1\mathrm{/}2}}(\mathrm{\Sigma }){\otimes }_{S_{q^{1\mathrm{/}2}}(\mathrm{\Sigma }{)}^{(N)}}\stackrel{~}{S_{q^{1\mathrm{/}2}}(\mathrm{\Sigma }{)}^{(N)}}$. Then we show the dimension of $\stackrel{~}{S_{q^{1\mathrm{/}2}}(\mathrm{\Sigma })}$ over $\stackrel{~}{S_{q^{1\mathrm{/}2}}(\mathrm{\Sigma }{)}^{(N)}}$ is $N^{3r(\mathrm{\Sigma })}$ where $r(\mathrm{\Sigma })$ equals to the number of boundary components of $\mathrm{\Sigma }$ minus the Euler characteristic of $\mathrm{\Sigma }$.