When the quantum parameter q1/2 is a root of unity of odd order. The
stated skein module Sq1/2​(M,N) has an
S1​(M,N)-module structure, where (M,N) is a marked
three manifold. We prove Sq1/2​(M,N) is a finitely generated
S1​(M,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
Sq1/2​(M,N) over S1​(M,N) when M is compact.
For a pb surface Σ, we use Sq1/2​(Σ)(N) to denote the
image of the Frobenius map when q1/2 is a root of unity of odd order N.
Then Sq1/2​(Σ)(N) lives in the center of the stated skein
algebra Sq1/2​(Σ). Let Sq1/2​(Σ)(N)​ be
the field of fractions of Sq1/2​(Σ)(N), and
Sq1/2​(Σ)​ be
Sq1/2​(Σ)⊗Sq1/2​(Σ)(N)​Sq1/2​(Σ)(N)​. Then we show the dimension of
Sq1/2​(Σ)​ over Sq1/2​(Σ)(N)​
is N3r(Σ) where r(Σ) equals to the number of boundary
components of Σ minus the Euler characteristic of Σ.Comment: 31 page