Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 113 / 4 / d / Solution

A Cartier divisor on an integral scheme is given by an open cover $(U_i)$ and rational functions $f_i\in K(X{)}^{\ast }$ such that $f_i/f_j\in {\mathcal{O}}_X^{\ast }(U_i\cap U_j)$. Two are linearly equivalent when their quotient is represented by one global rational function. The Cartier class group is the group of Cartier divisors modulo these principal divisors.

Let ${\mathcal{K}}_X^{\ast }$ be the sheaf of nonzero rational functions. Cartier divisors are the global sections of ${\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$, and the exact sequencegives a long exact cohomology sequence. On an integral scheme ${\mathcal{K}}_X^{\ast }$ is flasque: every nonempty restriction map is the identity on $K(X{)}^{\ast }$. Hence $H^1(X,{\mathcal{K}}_X^{\ast })=0$, and exactness gives

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