Simplicial Toric Varieties

Question

I am trying to solve an exercise in the well known book "Toric Varieties" by Cox, Little and Schenck:

Prop $4.2.7$: Let $X_\Sigma$ be the toric variety of the fan $\Sigma$. Then the following are equivalent:

a) Every Weil divisor on $X_\Sigma$ has a positive multiple that is Cartier.

b) $\operatorname{Pic}(X_\Sigma)$ has finite index in $\operatorname{Cl}(X_\Sigma)$.

c) $X_\Sigma$ is simplicial.

I am struggling with c) $\Rightarrow$ a). Could someone give me a hint?

Thank you!

Answer 1

For a simplicial cone its generators form a $\mathbb{Q}$-basis of the lattice, hence for any collection of integer labels on the generators there is a linear function on the cone that takes these values at the generators and rational values at other integral points. A multiple of this function is integral-valued on the lattice, hence corresponds to a Cartier divisor.