Finiteness and dimension of stated skein modules over Frobenius

Abstract

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 $Σ$, we use $S_{q^{1/2}}(Σ)^{(N)}$ to denote the image of the Frobenius map when $q^{1/2}$ is a root of unity of odd order $N$. Then $S_{q^{1/2}}(Σ)^{(N)}$ lives in the center of the stated skein algebra $S_{q^{1/2}}(Σ)$. Let $\widetilde{S_{q^{1/2}}(Σ)^{(N)}}$ be the field of fractions of $S_{q^{1/2}}(Σ)^{(N)}$, and $\widetilde{S_{q^{1/2}}(Σ)}$ be $S_{q^{1/2}}(Σ)\otimes_{S_{q^{1/2}}(Σ)^{(N)}} \widetilde{S_{q^{1/2}}(Σ)^{(N)}}$. Then we show the dimension of $\widetilde{S_{q^{1/2}}(Σ)}$ over $\widetilde{S_{q^{1/2}}(Σ)^{(N)}}$ is $N^{3r(Σ)}$ where $r(Σ)$ equals to the number of boundary components of $Σ$ minus the Euler characteristic of $Σ$.