I'm reading Dwyer and Fried's paper "Homology of free abelian covers, I". In it, they make the following claim, which I'm having trouble verifying.

Let $F$ be a field and $A = F[x_1^{\pm 1},\ldots,x_{\beta}^{\pm 1}]$ be a ring of Laurent polynomials over $F$. Let $M$ be a finitely-generated module over $A$. Recall that the support of $M$ is the set of all prime ideals $p$ of $A$ such that $M_p \neq 0$, or equivalently which satisfy $\text{ann}(M) \subset p$. The claim that I'm having trouble verifying is that $M$ is finite-dimensional over $F$ if and only if the support of $M$ consists of finitely many prime ideals.

Thanks for any help!