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For a scheme \(X\), the sheaf \(\mathcal K\) is the sheaf associated to the presheaf \(U\mapsto K(U)\), where \(K(U)\) is the total quotient ring of \(\mathcal O_X(U)\), i.e. \(\mathcal O_X(U)\) localized at the multiplicative system consisting of all non-zero divisors.
Definition
A Cartier divisor on a scheme \(X\) is a global section of the sheaf \(\mathcal K^*/\mathcal O^*\). More explicitly, a Cartier divisor \(C\) can be described by the data \((U_i, f_i)\) where \(\{U_i\}\) is an open cover of \(X\) and \(f_i \in \Gamma(U_i, \mathcal K^*)\) is an invertible global section of \(\mathcal K\) such that for each \(i,j\) the transition function \(f_i/f_j \in \Gamma(U_i\cap U_j, \mathcal O_X^*)\).
- We say that a Cartier divisor is principal if it is in the image of the map \(\Gamma(X,\mathcal K^*)\to \Gamma(X, \mathcal K^*/\mathcal O_X^*)\); i.e. if all the \(f_i\) are in fact the restrictions of a single \(f\in \Gamma(X, \mathcal K^*)\).
- We say that two Cartier divisors are linearly equivalent if their difference is principal; i.e. if there is some function \(g\) such that \(g|_{U_i\cap U_i'} = f_i/f_i'\). Note that even though the group operation on \(\mathcal K^*/\mathcal O_X^*\) is multiplicative, we use additive language to maintain an analogy to Weil divisors.
- We say that a Cartier divisor is effective if it is represented by \(\{(U_i, f_i)\}\) where for each \(i\) \(f_i\in \Gamma(U_i, \mathcal O_{U_i})\).
The set of linear equivalence classes of Cartier divisors on \(X\) is denoted \(\operatorname{CaCl}(X)\) and is a group.
The Weil divisor associated to a Cartier divisor \(C = \{(U_i, f_i)\}\) should be thought of as the locus \(D\) of \(X\)where the \(f_i\) has either a pole or a zero, split up into irreducibles \(D = Y_1\cup...\cup Y_m\), and then weighted by the order of vanishing of the \(f_i\) along those irreducible subsets.