= Cartier divisor
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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>.<The Stacks project>[tag02AR, Definition111.49.1, parts6–7: total quotient rings and Cartier divisors]{ref}
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