Finiteness and dimension of stated skein modules over Frobenius

When the quantum parameter [Formula: see text] is a root of unity of odd order, the stated skein module [Formula: see text] has an [Formula: see text]-module structure, where [Formula: see text] is a marked 3-manifold. We prove [Formula: see text] is a finitely generated [Formula: see text]-module when [Formula: see text] is compact, which furthermore shows the reduced stated skein module for the compact marked 3-manifold is finite-dimensional over [Formula: see text]. We also give an upper bound for the dimension of [Formula: see text] over [Formula: see text] when [Formula: see text] is compact. For a punctured bordered surface [Formula: see text], we use [Formula: see text] to denote the image of the Frobenius map when [Formula: see text] is a root of unity of odd order [Formula: see text]. Then [Formula: see text] lives in the center of the stated skein algebra [Formula: see text]. Let [Formula: see text] be the field of fractions of [Formula: see text], and [Formula: see text] be [Formula: see text]. Then we show the dimension of [Formula: see text] over [Formula: see text] is [Formula: see text] where [Formula: see text] equals to the number of boundary components of [Formula: see text] minus the Euler characteristic of [Formula: see text].