residue field

{{Short description|Field arising from a quotient ring by a maximal ideal}}

In mathematics, the residue field is a basic construction in commutative algebra. If $R$ is a commutative ring and $\mathfrak{m}$ is a maximal ideal, then the residue field is the quotient ring $k=R/\mathfrak{m}$ , which is a field.{{cite book

| last1 = Dummit

| first1 = D. S.

| last2 = Foote

| first2 = R.

| title = Abstract Algebra

| publisher = Wiley

| year = 2004

| edition = 3

| isbn = 9780471433347

}} Frequently, $R$ is a local ring and $\mathfrak{m}$ 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 $k(x)$ .{{cite book

| author = David Mumford

| author-link = David Mumford

| year = 1999

| title = The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians

| series = Lecture Notes in Mathematics

| volume = 1358

| edition = 2nd

| publisher = Springer-Verlag

| doi = 10.1007/b62130

| isbn = 3-540-63293-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.{{clarify|date=February 2015}}

Definition

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

Now suppose that $X$ is a scheme and $x$ is a point of $X$ . By the definition of a scheme, we may find an affine neighbourhood $\mathcal{U} = \text{Spec}(A)$ of $x$ , with some commutative ring $A$ . Considered in the neighbourhood $\mathcal{U}$ , the point $x$ corresponds to a prime ideal $\mathfrak{p} \subseteq A$ (see Zariski topology). The local ring of $X$ at $x$ is by definition the localization $A_{\mathfrak{p}}$ of $A$ by $A\setminus \mathfrak{p}$ , and $A_{\mathfrak{p}}$ has maximal ideal $\mathfrak{m}=\mathfrak{p} A_{\mathfrak{p}}$ . Applying the construction above, we obtain the residue field of the point $x$ :

: $k(x) := A_{\mathfrak{p}}/\mathfrak{p} A_{\mathfrak{p}}$ .

Since localization is exact, $k(x)$ is the field of fractions of $A/\mathfrak p$ (which is an integral domain as $\mathfrak p$ is a prime ideal).{{Matsumura CA}}, 1.K One can prove that this definition does not depend on the choice of the affine neighbourhood $\mathcal{U}$ .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.

A point is called $\color{blue}k$ -rational for a certain field $k$ , if $k(x)=k$ .Görtz, Ulrich and Wedhorn, Torsten. Algebraic Geometry: Part 1: Schemes (2010) Vieweg+Teubner Verlag.

Example

Consider the affine line $\mathbb{A}^1(k)=\operatorname{Spec}(k[t])$ 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 from the irreducible polynomials of degree greater than 1. For example if $k=\mathbb{R}$ , then the prime ideals generated by quadratic irreducible polynomials (such as $x^2+1$ ) all have residue field isomorphic to $\mathbb{C}$ .

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 $\operatorname{Spec}(K) \rightarrow X$ , $K$ 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.

residue field

Definition

Example

Properties

See also

References

Further reading