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. 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 $k(x)$. 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 $\mathfrak{𝔪}$. 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 ${𝒰}=\text{Spec}(A)$ of $x$, with some commutative ring $A$. Considered in the neighbourhood ${𝒰}$, 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). 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 $k(x)=k$.

Example

Consider the affine line ${\mathbb{𝔸}}^1(k)=\mathrm{Spec}(k[t])$ 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 $\mathrm{Spec}(K)\to X$, $K$ 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

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