The localized skein algebra is Frobenius

摘要

When $A$ in the Kauffman bracket skein relation is set equal to a primitive $n^{ext{th}}$ root of unity $zeta$ with $n$ not divisible by $4$, the Kauffman bracket skein algebra $K_{zeta}(F)$ of a finite-type surface $F$ is a ring extension of the $operatorname{SL}_2mathbb{C}$--character ring of the fundamental group of $F!$. We localize by inverting the nonzero characters to get an algebra $S^{-1}K_{zeta}(F)$ over the function field of the corresponding character variety. We prove that if $F$ is noncompact, the algebra $S^{-1}K_{zeta}(F)$ is a symmetric Frobenius algebra. Along the way we prove $K(F)$ is finitely generated, $K_{zeta}(F)$ is a finite-rank module over the coordinate ring of the corresponding character variety, and learn to compute the trace that makes the algebra Frobenius.