The line bundle associated to a Cartier divisor (definition)
Definition
Let \(D\) be a Cartier divisor on a scheme \(X\) represented by \(\{(U_i, f_i)\}\). Define \(\mathcal O_X(D)\) to be the sub \(\mathcal O_X\)-module of the sheaf \(\mathcal K\) of total quotient rings locally generated by \(f_i^{-1}\) on \(U_i\); that is,
\begin{align*} \mathcal O_X(D)(U_i) = \frac{\mathcal O_X(U_i)}{f_i}. \end{align*}We call this the line bundle associated to \(D\), terminology justified by the following proposition.
Cartier divisors up to linear equivalence uniquely determine line bundles up to isomorphism
Proposition
Let \(X\) be a scheme. Then
- For any Cartier divisor \(D\) on \(X\) the scheme \(\mathcal O_X(D)\) is an invertible sheaf (line bundle) on \(X\). The association \(D\mapsto \mathcal O_X(D)\) gives a one-to-one correspondence between Cartier divisors on \(X\) and invertible subsheaves of \(\mathcal K\).
- \(\mathcal O_X(D_1 - D_2) = \mathcal O_X(D_1)\otimes \mathcal O_X(D_2)^{-1}\)
- \(D_1\) is linearly equivalent to \(D_2\) if and only if \(\mathcal O_X(D_1)\cong \mathcal O_X(D_2)\) as abstract invertible sheaves.
In particular, the association \(D\mapsto \mathcal O_X(D)\) is an injective homomorphism \(\operatorname{CaCl}(X)\hookrightarrow \Pic(X)\).
There exist pathological situations where not every line bundle is given by the data of a Cartier divisor, but in all typical situations the above homomorphism is an isomorphism.