I am trying to understand the following exercise:
Let $S \subset M$ be a sub-semigroup of a lattice $M$. If $S$ is saturated in $M$ (meaning that if $p \cdot m \in S$ for $p \in \mathbb{Z}_{\geq 1}$ and $m \in M$, then $m \in S$), then $\mathbb{C}[S]$ is integrally closed.
For context, this comes from page 30 of Fulton's 'Introduction to Toric Varieties'. I was able to find it here as Lemma 1.12, but the proof uses some kind of torus action on the ring $\mathbb{C}[M]$, instead of its spectrum, and I'm not sure how that works, or why it would fix $\mathbb{C}[S]$.
The way to start I think is to first note that $$\mathbb{C}[M] = \mathbb{C}[\chi^{\pm e_1}, \dots, \chi^{\pm e_{n}}] \cong \mathbb{C}[x_1^{\pm 1}, \dots, x_n^{\pm 1}]$$ is integrally closed, so the integral closure of $\mathbb{C}[S]$ will be a subring of $\mathbb{C}[M]$, so it suffices to find the integral closure of $\mathbb{C}[S]$ in $\mathbb{C}[M]$. From here I was pretty stuck, since arguing directly gets messy, since general terms of $\mathbb{C}[M]$ will be $k$-linear combinations of $\chi^{u}$. I think this is why the linked argument uses some kind of group action argument.
Anyway, any help in proving this fact is greatly appreciated!
Thanks!
