Divisor (algebraic Geometry) - Cartier Divisor

A Cartier divisor can be represented by an open cover by affine sets, and a collection of rational functions defined on . The functions must be compatible in this sense: on the intersection of two sets in the cover, the quotient of the corresponding rational functions should be regular and invertible. A Cartier divisor is said to be effective if these can be chosen to be regular functions, and in this case the Cartier divisor defines an associated subvariety of codimension 1 by forming the ideal sheaf generated locally by the .

The notion can be described more conceptually with the function field. For each affine open subset U, define M′(U) to be the total quotient ring of O_X(U). Because the affine open subsets form a basis for the topology on X, this defines a presheaf on X. (This is not the same as taking the total quotient ring of O_X(U) for arbitrary U, since that does not define a presheaf.) The sheaf M_X of rational functions on X is the sheaf associated to the presheaf M′, and the quotient sheaf M_X* / O_X* is the sheaf of local Cartier divisors.

A Cartier divisor is a global section of the quotient sheaf M_X*/O_X*. We have the exact sequence, so, applying the global section functor gives the exact sequence .

A Cartier divisor is said to be principal if it is in the range of the morphism, that is, if it is the class of a global rational function.