Cartier divisor

A Cartier divisor is locally represented by nonzero rational functions whose ratios on overlaps are regular units. It is principal when one global rational function represents all local data.

For a Cartier divisor $D$, the associated line bundle $\mathcal{O}(D)$ has local sections given by meromorphic functions $f$ satisfying $(f)+D\ge 0$. Its canonical meromorphic section has divisor $D$.

The Cartier class group is the group of Cartier divisors modulo principal Cartier divisors. On an integral scheme it is naturally $H^1(X,{\mathcal{O}}_X^{\ast })$.