Lemma
Let \(X\) be a noetherian scheme. To each Cartier divisor on \(X\) we can associate a unique Weil divisor. This association respects linear equivalence and thus gives us an injective morphism \(\operatorname{CaCl}(X) \hookrightarrow \Cl(X)\) from the group of Cartier divisors to the divisor class group of \(X\). When \(X\) is additionally integral and separated, this homomorphism is an isomorphim.
Proof
Cartier to Weil. Fix a Cartier divisor \(C\) on \(X\) given by local data \(\{(U_i, f_i)\}\). Given a prime divisor \(Y\subset X\), there is some \(U_Y\) in the collection \(\{U_i\}\) such that \(U_Y\cap Y\neq \emptyset\). Choosing one such \(U_Y\) for each \(Y\), we define the Weil divisor \(D\) associated to \(C\) as follows:
\begin{align*} C\mapsto \sum_{\substack{Y \subset X \\ \text{a prime divisor} }} v_Y(f_Y)[Y] = D \end{align*}- This is well defined because for any other \(i\) such that \(U_i \cap Y \neq \emptyset\), \(f_Y/f_i\) is invertible, and hence \(v_Y(f_Y/f_i) = 0\) and \(v_Y(f_Y) = v_Y(f_i).\) The sum defining \(D\) is finite because \(X\) is noetherian.
- This respects linear equivalence for if \((U_i, f_i)\) is principal, then there is some \(f \in \Gamma(X, \mathcal K^*)\) such that \(f_i = f|_{U_i}\) and thus
Weil to Cartier. Now suppose \(X\) is integral and separated and let \(D\) be a Weil divisor. By integrality, the sheaf \(\mathcal K\) is just the constant sheaf equal to the function field \(K\) of \(X\). Then at each point \(x\in X\), \(D_x\) is a Weil divisor on the local scheme \(\Spec \mathcal O_{X,x}\). Since \(\mathcal O_{X,x}\) is a UFD, every divisor is principal and hence \(D_x = \operatorname{div}(f_x)\) for some \(f_x\in K\).
The divisor \(D\) and \(\operatorname{div}(f_x)\) now differ only along prime divisors which do not contain \(x\), and only finitely many of these prime divisors have zero coefficient in either \(\operatorname{div}(f_x)\) or \(D\). Taking \(U_x\) to be the complement of these divisors, we get that the restriction of \(\operatorname{div}(f_x)\) and \(D\) to \(U_x\) are equal. Repeating this proceedure for one point \(x\) on each of the finitely many prime divisors forming the complement of \(U_x\) produces a Cartier divisor \(C\).
It remains to check that the quotients \(f_x/f_{x'}\) are in \(\mathcal O_X^*\) and that these two maps are inverses of each other.
Note: a Weil divisor \(D\) is said to be locally principal if \(X\) can be covered with open sets \(U\) such that for each \(U\) \(D|_U\) is a principal divisor. The above proof shows that, when we remove the integral and separated hypotheses, the image of \(\operatorname{CaCl}(X)\) in \(\Cl(X)\) is exactly the locally principal divisors.