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(x). 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.

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(A), with A some commutative ring. Considered in the neighbourhood U, the point x corresponds to a prime ideal p ⊂ A (see Zariski topology). The local ring of X in x is by definition the localization R = A_p, with the maximal ideal m = p·A_p. 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(x) ⊂ K.

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