Cartier divisor

Cartier divisors

Cartier divisors also have a sheaf-theoretic description. A fractional ideal sheaf is a sub-${\displaystyle {\mathcal {O}}_{X}}$-module of ${\displaystyle {\mathcal {M}}_{X}.}$ A fractional ideal sheaf J is invertible if, for each x in X, there exists an open neighborhood U of x on which the restriction of J to U is equal to ${\displaystyle {\mathcal {O}}_{U}\cdot f,}$ where ${\displaystyle f\in {\mathcal {M}}_{X}^{\times }(U)}$ and the product is taken in ${\displaystyle {\mathcal {M}}_{X}.}$ Each Cartier divisor defines an invertible fractional ideal sheaf using the description of the Cartier divisor as a collection ${\displaystyle \{(U_{i},f_{i})\},}$ and conversely, invertible fractional ideal sheaves define Cartier divisors. If the Cartier divisor is denoted D, then the corresponding fractional ideal sheaf is denoted ${\displaystyle {\mathcal {O}}(D)}$ or L(D).