Residue field

In mathematics, the residue field is a basic construction in commutative algebra. If $R$ is a commutative ring and $\mathfrak$ is a maximal ideal, then the residue field is the quotient ring $k=R/\mathfrak$, which is a field. Frequently, $R$ is a local ring and $\mathfrak$ 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 $\mathfrak$. Then the residue field is the quotient ring $R/\mathfrak$.

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 $\mathcal = \text(A)$ of $x$, with some commutative ring $A$. Considered in the neighbourhood $\mathcal$, the point $x$ corresponds to a prime ideal $\mathfrak \subseteq A$ (see Zariski topology). The ''local ring'' of $X$ at $x$ is by definition the localization $A_$ of $A$ by $A\setminus \mathfrak$, and $A_$ has maximal ideal $\mathfrak=\mathfrak A_$. Applying the construction above, we obtain the residue field of the point $x$:

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

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 $\mathcal$.

A point is called $\colork$-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^1(k)=\operatorname(k 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-a)k \cong k
*k \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 $k=\mathbb$, then the prime ideals generated by quadratic irreducible polynomials (such as $x^2+1$) all have residue field isomorphic to $\mathbb$.

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 $\operatorname(K) \rightarrow X$, $K$ 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

References

Further reading

* {{Citation , last1=Hartshorne , first1=Robin , author1-link = Robin Hartshorne , title=Algebraic Geometry , publisher=Springer-Verlag , location=Berlin, New York , isbn=978-0-387-90244-9 , mr=0463157 , year=1977, section II.2

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