Residue field


In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field. Frequently, R is a local ring and m is then its unique maximal ideal.
This construction is applied in algebraic geometry, where to every point x of a scheme X one associates its residue field k. One can say a little loosely that the residue field of a point of an abstract algebraic variety is the 'natural domain' for the coordinates of the point.

Definition

Suppose that R is a commutative local ring, with the maximal ideal m. Then the residue field is the quotient ring R/m.
Now suppose that X is a scheme and x is a point of X. By the definition of scheme, we may find an affine neighbourhood U = Spec, with A some commutative ring. Considered in the neighbourhood U, the point x corresponds to a prime ideal p ⊂ A. The local ring of X in x is by definition the localization R = Ap, with the maximal ideal m = p·Ap. Applying the construction above, we obtain the residue field of the point x :
One can prove that this definition does not depend on the choice of the affine neighbourhood U.
A point is called K-rational for a certain field K, if k = K.

Example

Consider the affine line A1 = Spec over a field k. If k is algebraically closed, there are exactly two types of prime ideals, namely
The residue fields are
If k is not algebraically closed, then more types arise, for example if k = R, then the prime ideal has residue field isomorphic to C.

Properties