Residue field explained

In mathematics, the residue field is a basic construction in commutative algebra. If

R is a commutative ring and ak{m} is a maximal ideal, then the residue field is the quotient ring k = R/ ak{m} , which is a field.^[1] Frequently, R is a local ring and ak{m} is then its unique maximal ideal.

In abstract algebra, the splitting field of a polynomial is constructed using residue fields. Residue fields also 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 maximal ideal ak{m} . Then the residue field is the quotient ring R/ ak{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 l{U} = Spec (A) of x , with some commutative ring A . Considered in the neighbourhood l{U} , the point x corresponds to a prime ideal ak{p} \subseteq A (see Zariski topology). The local ring of X at x is by definition the localization A _{ak{p} } of A by A\setminus ak{p} , and A _{ak{p} } has maximal ideal ak{m} = ak{p} A _{ak{p} }. Applying the construction above, we obtain the residue field of the point x :k(x) : = A _{ak{p} }/\mathfrak A_ .

One can prove that this definition does not depend on the choice of the affine neighbourhood

l{U} .^[3]

A point is called

\color{blue}k -rational for a certain field

k , if k(x) = k .^[4]

Example

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 \in k ( 0 ) , the zero-ideal.

The residue fields are

k[t] _{(t-a)} /(t - a)k[t] _{(t-a)} \cong k 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 (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 \in 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

Further reading

• , section II.2

Notes and References

1. Book: Dummit. D. S.. Foote. R.. Abstract Algebra. Wiley. 2004. 3. 9780471433347 .
2. Book: David Mumford . David Mumford . 1999 . The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians . Lecture Notes in Mathematics . 1358 . 2nd . Springer-Verlag . 10.1007/b62130 . 3-540-63293-X.
3. 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.
4. [Ulrich Görtz|Görtz, Ulrich]