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.^{[clarification needed]}

Definition

Suppose that R is a commutative local ring, with maximal ideal m. Then the residue field is the quotient ring R/m.

Now suppose that X is a

A_p of A by A \ p, and A_p has maximal ideal m = p·A_p. Applying the construction above, we obtain the 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

, there are exactly two types of prime ideals, namely

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

The residue fields are

• ${\displaystyle k[t]_{(t-a)}/(t-a)k[t]_{(t-a)}\cong k}$
• ${\displaystyle k[t]_{(0)}\cong k(t)}$, 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 (x² + 1) has residue field isomorphic to C.

Properties

• For a scheme locally of
  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.

See also

• Arithmetic zeta function

References

Further reading

• MR 0463157
  , section II.2