# 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.^{[1]} 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*).^{[2]} 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(*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*. Applying the construction above, we obtain the

_{p}**residue field of the point**:

*x**k*(*x*) :=*A*_{p}/*p*·*A*_{p}.

One can prove that this definition does not depend on the choice of the affine neighbourhood *U*.^{[3]}

A point is called *K*-rational for a certain field *K*, if *k*(*x*) ⊂ *K*.^{[4]}

## Example

Consider the affine line **A**^{1}(*k*) = Spec(*k*[*t*]) over a field *k*. If *k* is algebraically closed, there are exactly two types of prime ideals, namely

- (
*t*−*a*),*a*∈*k* - (0), the zero-ideal.

The residue fields are

- , the function field over
*k*in one variable.

If *k* is not algebraically closed, then more types arise, for example if *k* = **R**, then the prime ideal (*x*^{2} + 1) has residue field isomorphic to **C**.

## Properties

- For a scheme locally of finite type over a field
*k*, a point*x*is closed if and only if*k*(*x*) is a finite extension of the base field*k*. This is a geometric formulation of Hilbert's Nullstellensatz. In the above example, the points of the first kind are closed, having residue field*k*, whereas the second point is the generic point, having transcendence degree 1 over*k*. - A morphism Spec(
*K*) →*X*,*K*some field, is equivalent to giving a point*x*∈*X*and an extension*K*/*k*(*x*). - The dimension of a scheme of finite type over a field is equal to the transcendence degree of the residue field of the generic point.

## References

- ↑ Dummit, D. S.; Foote, R. (2004).
*Abstract Algebra*(3 ed.). Wiley. ISBN 9780471433347. - ↑ David Mumford (1999).
*The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians*(2nd ed.). Springer-Verlag. doi:10.1007/b62130. ISBN 3-540-63293-X. - ↑ Intuitively, the residue field of a point is a local invariant. Axioms of schemes are set up in such a way as to assure the compatibility between various affine open neighborhoods of a point, which implies the statement.
- ↑ Görtz, Ulrich and Wedhorn, Torsten.
*Algebraic Geometry: Part 1: Schemes*(2010) Vieweg+Teubner Verlag.

## Further reading

- Hartshorne, Robin (1977),
*Algebraic Geometry*, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, section II.2