Cartier divisor

Summary

A Cartier divisor on a scheme is a hypersurface described locally by one equation. The local equations are given up to multiplication by invertible functions. Each Cartier divisor determines an invertible sheaf, and on an integral scheme every invertible sheaf arises this way. So the classes of Cartier divisors give the Picard group of an integral scheme. On a Noetherian integral separated scheme whose local rings are factorial, Cartier divisors are the same as Weil divisors. Cartier divisors are the right kind of divisor on singular schemes, where Weil divisors and line bundles no longer match.