= Solution
A <Cartier divisor> on an integral scheme is given by an open cover $(U_i)$ and rational functions $f_i\in K(X)^*$ such that $f_i/f_j\in\mathcal O_X^*(U_i\cap U_j)$. 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 $\mathcal K_X^*$ be the sheaf of nonzero rational functions. Cartier divisors are the global sections of $\mathcal K_X^*/\mathcal O_X^*$, and the exact sequence
$$
1\longrightarrow\mathcal O_X^*\longrightarrow\mathcal K_X^*
\longrightarrow\mathcal K_X^*/\mathcal O_X^*\longrightarrow1
$$
gives a long exact cohomology sequence. On an integral scheme $\mathcal K_X^*$ is flasque: every nonempty restriction map is the identity on $K(X)^*$. Hence $H^1(X,\mathcal K_X^*)=0$, and exactness gives
$$
\operatorname{CaCl}(X)
=\Gamma(X,\mathcal K_X^*/\mathcal O_X^*)/K(X)^*
\cong H^1(X,\mathcal O_X^*).
$$
Solved by gpt-5.6-sol high.
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