= Divisor class groups of products of projective spaces
{title2=$\operatorname{Cl}(\mathbb P^n\times\mathbb P^m)=\mathbb Z^2$}
For positive $n,m$, $\operatorname{Cl}(\mathbb P^n\times\mathbb P^m)=\mathbb Z^2$, generated freely by the pullbacks of coordinate hyperplanes. Restrict a divisor to the standard affine chart, where <unique factorization> makes it principal, and subtract this <principal Weil divisor> to leave only the two boundary hyperplanes. Any principal relation between them would be given by a rational function whose divisor vanishes on the chart; <unique factorization> then makes that function a constant, proving independence. The same argument gives $\operatorname{Cl}(\mathbb P^r)=\mathbb Z$.
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