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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!

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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.

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  • $\begingroup$ Hey, sorry for the late reply. I don't really understand how to interpret Divisors as linear functions on the cone... In the book they rather give the advice to look at the map sending the canonical basis vectors of $\mathbb{Z}^{r}$ to the primitive generators and it's dual under the assumption, that the primitive generators are linearly independent. $\endgroup$ – Vasco1008 Aug 9 '20 at 14:40

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