1

I am solving a problem in W. Fulton, Introduction to Toric Varieties, section 2.1, page 30.

Let $M$ be a lattice and $S$ one of its affine sub-semigroup, then the normalization of $\mathbb{C}[S]$ is $\mathbb{C}[\overline{S}]$ where $\overline{S} = \left \{p \in M \mid \exists k \in \mathbb{N}^*: kp \in S \right \}$ is the saturation of $S$.

Firstly, it is obviously that $\overline{S}$ is saturated then it is well-known that $\mathbb{C}[\overline{S}]$ is integrally closed. The extension $\mathbb{C}[S] \to \mathbb{C}[\overline{S}]$ is integral because every "elementary" character $\chi^p \in \mathbb{C}[\overline{S}]$ is a root of the polynomial $X^k - \chi^{kp} \in \mathbb{C}[S][X]$ for $kp \in S$. So to conclude that $\mathbb{C}[\overline{S}]$ is the integral closure of $\mathbb{C}[S]$ we need to show that the fields of fractions coincide. Because a field is closed under taking sum or inverse, we just need to prove that every element of form $\frac{\chi^p}{1} \in \mathrm{Frac}(\mathbb{C}[\overline{S}])$ is actually in $\mathrm{Frac}(\mathbb{C}[S])$ which is the point I am stuck at. Thank you in advance for any hint.

user26857
  • 52,094
Alexey Do
  • 2,109
  • 1
    I'm not sure if this is true. If $S = 2 \mathbb N \subset \mathbb Z$, then $\mathbb C[S] = \mathbb C[x^2]$, which is already integrally closed, so $\mathbb C[\overline S] = \mathbb C[x]$ cannot be its integral closure. Or is that somehow not allowed? – red_trumpet Jan 28 '22 at 23:02
  • @red_trumpet $\mathbb{C}[x^2]$ is not integrally closed as $a^2-x^2=0$ has $x$ as a root which is not in $\mathbb{C}[x^2]$. I do not know if this is helpful or not but there is another way to describe $\overline{S}$, $\overline{S}=\mathrm{Cone}(S)\cap M$ (this is highly nontrivial to prove!) You probably want to have a look at D.Cox, Toric Varieties, proposition 1.3.8 – Alexey Do Jan 29 '22 at 12:59
  • 1
    $\mathbb C[x^2]$ is not integrally closed in $\mathbb C[x]$, but it is integrally closed (in its field of fractions, which actually differs from the field of fractions of $\mathbb C[x]$). – user26857 Jan 29 '22 at 13:42
  • @user26857 yes, sorry, that's my mistake, so perhaps the question is wrong and we should add something to the hypothesis, like $\mathbb{C}[\overline{S}]$ is the integral closure of $\mathbb{C}[S]$ in $\mathbb{C}[M]$. How do you think? – Alexey Do Jan 29 '22 at 14:02

0 Answers0