Divisor (algebraic Geometry) - From Cartier Divisors To Weil Divisor

From Cartier Divisors To Weil Divisor

There is a natural homomorphism from the group of Cartier divisors to that of Weil divisors, which is an isomorphism for integral separated Noetherian schemes provided that all local rings are unique factorization domains.

In general Cartier behave better than Weil divisors when the variety has singular points.

An example of a surface on which the two concepts differ is a cone, i.e. a singular quadric. At the (unique) singular point, the vertex of the cone, a single line drawn on the cone is a Weil divisor — but is not a Cartier divisor.

The divisor appellation is part of the history of the subject, going back to the Dedekind–Weber work which in effect showed the relevance of Dedekind domains to the case of algebraic curves. In that case the free abelian group on the points of the curve is closely related to the fractional ideal theory.