a Cartier divisor

On a curve or surface you often want to specify a 'codimension-one locus with multiplicities' — where a function has zeros or poles, counted with their orders. There are two ways to make this precise, and they correspond to two flavours of divisor. A Weil divisor is the direct approach: a formal integer combination of codimension-one subvarieties. A Cartier divisor is the more flexible, scheme-friendly approach: data of local equations that, on overlaps, differ only by invertible functions. The Cartier version is the one that works on any scheme and ties directly to line bundles.

Precisely, a Cartier divisor on a scheme X is given by an open cover {U_i} together with, on each U_i, a nonzero rational (meromorphic) function f_i, subject to the compatibility that on every overlap U_i intersect U_j the ratio f_i / f_j is a unit — an invertible regular function — there. The f_i are the local equations of the divisor; the compatibility says the divisor 'is the same locus' from chart to chart, with the choice of defining equation allowed to differ by something that never vanishes. A Cartier divisor is effective if every f_i is a genuine regular function (so it really cuts out a codimension-one subscheme). Two Cartier divisors are linearly equivalent if they differ by the divisor of a single global rational function. The whole point is the dictionary with line bundles: a Cartier divisor D determines an invertible sheaf O_X(D) — glue trivial line bundles on the U_i using the units f_i/f_j as transition functions — and this gives a homomorphism from Cartier divisors to Pic(X) that is an isomorphism modulo principal divisors. So Cartier divisors modulo linear equivalence = Pic(X).

Cartier divisors are the right notion for Riemann-Roch and intersection theory on schemes, precisely because every Cartier divisor is locally principal and therefore always corresponds to a line bundle. Honest cautions about the Weil/Cartier distinction, which is the crux. On a smooth variety (more generally a locally factorial scheme) Weil and Cartier divisors agree, and you may pass freely between 'codimension-one cycle' and 'local equations'. But on a singular scheme they GENUINELY differ: a Weil divisor through a singular point need not be locally principal, hence is not Cartier and corresponds to no line bundle. The standard example is the cone Spec k[x, y, z]/(xy - z^2): a ruling line through the vertex is a Weil divisor that is not Cartier (twice it is Cartier, but it itself is not). So 'divisor = line bundle' is true only via the Cartier side, and only the Cartier-to-Pic map is always defined; the Weil class group can be strictly larger.