Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 113 / 2 / c

For a Noetherian separated integral scheme regular in codimension one, a Weil divisor is a finite integer combination of integral codimension-one closed subschemes. Principal divisors are the valuation divisors of nonzero rational functions, and the divisor class group is

For an open immersion $U\subseteq X$, restriction induces a surjectionevery prime divisor of $U$ closes to a prime divisor of $X$, while divisors supported in $X\setminus U$ form the kernel. Therefore $\mathrm{Cl}(X)=0$ implies $\mathrm{Cl}(U)=0$.

Affine schemes need not have trivial class group. For example,is a normal affine surface with $\mathrm{Cl}(X)\cong \mathbb{Z}/2\mathbb{Z}$. The height-one prime $(x,z)$ represents the nonzero class: twice this divisor is the principal divisor of $x$, but the divisor itself is not principal.