# What is wrong with this proof of $\operatorname{Cl}(X_\Sigma) \simeq \mathbb{Z}^{r-n}$?

This question is a bit urgent, as I have written this in my Master's thesis and I have to submit it next week!

Proposition: Let $$X_\Sigma$$ be a toric variety without torus factors, meaning that the minimal generator of the rays $$u_\rho$$ generate the whole lattice $$N$$, then with $$r=|\Sigma(1)|$$ (the number of one-dimensional cones of the fan) we get the isomorphism $$\operatorname{Cl}(X_\Sigma)\simeq \mathbb{Z}^{r-n}$$ Proof. We take the exact sequence from CLS ("Toric varieties" from Cox, Little, Schenck) Theorem 4.1.3 (p. 172) together with the remark directly above: $$0 \longrightarrow M\simeq \mathbb{Z}^n \longrightarrow \operatorname{Div}_{T_N}(X_\Sigma)\simeq \bigoplus_{\rho\in\Sigma(1)} \mathbb{Z} D_\rho\longrightarrow \operatorname{Cl}(X_\Sigma)\longrightarrow 0.$$ This simplifies to the short exact sequence $$0 \longrightarrow \mathbb{Z}^n \longrightarrow \mathbb{Z}^{r} \longrightarrow \operatorname{Cl}(X_\Sigma)\longrightarrow 0$$ and with the homomorphism theorem, i.e. for any homomorphism $$f:G\to H$$ we have $$G/\ker f \simeq \operatorname{im}f$$ which yields the claim. $$\square$$

I don't think this is true, as CLS (and other resources) add a finite group to the divisor class group. If $$\Sigma$$ contains an $$n$$-dimensional cone, I have found out that $$\operatorname{Pic}(X_\Sigma)\simeq \mathbb{Z}^{r-n}$$, but only if $$\Sigma$$ is smooth, we have an equality with $$\operatorname{Cl}(X_\Sigma)$$.

Where is my logical error, or can someone provide a counterexample? This proof seems to be too simple that nobody else would already have found it.

• The quotient depends on the details of the map. That is, the quotient of $\mathbb{Z}^{r}$ by $\mathbb{Z}^{n}$ simply need not be $\mathbb{Z}^{r-n}$. A simple example is given by the case $n=r=1$ and the map "multiply by $2$". Nov 12, 2021 at 20:40
• Oh shit, you're right. Is there any criterion, how one can make sure that this can't happen? Nov 12, 2021 at 20:52
• That's a special case of the structure theorem for finitely generated modules over a PID (the words "Smith Normal Form" might ring a bell). As for criteria, smoothness is one criterion, but I suppose that's not satisfactory for some reason? Nov 12, 2021 at 20:57
• If I have time tonight then I'll find a decent toric example to actually illustrate the phenomenon. If not, then I encourage you to find one (either after you've submitted your thesis, or as a good example to include). Nov 12, 2021 at 21:30
• Now you are really asking other questions, but I can direct you to Propositio 4.2.6 of CLS on page 179 and its proof on the next page. Also, it is perhaps worth noting that "Smith normal form" gets a mention on page 173. Nov 12, 2021 at 21:42