 Cartier divisor

On an integral scheme $X$, a Cartier divisor is represented on an open cover $U_i$ by nonzero rational functions $f_i$, with $f_i/f_j$ a regular unit on every overlap. Changing the local equation by a regular unit leaves the divisor unchanged. A principal Cartier divisor uses one global nonzero rational function on all the charts.

The transition ratios glue the local rank-one modules $f_i^{-1}{\mathcal{O}}_{U_i}$ into the divisor line bundle ${\mathcal{O}}_X(D)$. If each $f_i$ is regular, it cuts out an effective Cartier divisor. Allowing poles gives signed divisor data.

On the projective line, the coordinate $t$ defines the principal Cartier divisor $[0]-[\infty ]$: it has one simple zero at zero and one simple pole at infinity. For a general scheme, the analogous definition uses invertible elements of the sheaf of total quotient rings, so nonzero elements must be distinguished from non-zero-divisors.^[ref]