Residue field

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 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 $x$ :

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

One can prove that this definition does not depend on the choice of the affine neighbourhood ${\mathcal {U}}$ .

A point is called $\color {blue}k$-rational for a certain field $k$, if $k(x)=k$.

Example

Consider the affine line $\mathbb {A} ^{1}(k)={\text{Spec}}(k[t])$ 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 $k=\mathbb {R}$, then the prime ideal $(x^{2}+1)$ has 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 ${\text{Spec}}(K)\rightarrow X$ , $K$ 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.

See also

• Arithmetic zeta function

Further reading

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