mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the residue field is a basic construction in

commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...

. If ''R'' is a

commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...

and ''m'' is a

maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...

, then the residue field is the

quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...

''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a

local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebrai ...

and ''m'' is then its unique maximal ideal. This construction is applied in

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, where to every point ''x'' of a

scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...

''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 Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...

is the 'natural domain' for the coordinates of the point.

Definition

Suppose that ''R'' is a commutative

, with maximal ideal ''m''. Then the residue field is the quotient ring ''R''/''m''. Now suppose that ''X'' is a

and ''x'' is a point of ''X''. By the definition of scheme, we may find an affine neighbourhood ''U'' = Spec(''A''), with ''A'' some

. Considered in the neighbourhood ''U'', the point ''x'' corresponds to a

prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...

''p'' ⊆ ''A'' (see

Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...

). The ''

'' of ''X'' in ''x'' is by definition the localization ''R'' = ''A_p'', with the maximal ideal ''m'' = ''p·A_p''. Applying the construction above, we obtain the residue field of the point ''x'' : :''k''(''x'') := ''A''_''p'' / ''p''·''A''_''p''. 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''. Görtz, Ulrich and Wedhorn, Torsten. ''Algebraic Geometry: Part 1: Schemes'' (2010) Vieweg+Teubner Verlag.

Example

Consider the

affine line In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties rela ...

A¹(''k'') = Spec(''k'' 't'' over a field ''k''. If ''k'' is

algebraically closed In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...

, there are exactly two types of prime ideals, namely *(''t'' − ''a''), ''a'' ∈ ''k'' *(0), the zero-ideal. The residue fields are *

\cong k

\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 mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ...

. In the above example, the points of the first kind are closed, having residue field ''k'', whereas the second point is the

generic point In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic g ...

, having

transcendence degree In abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ...

1 over ''k''. * A morphism Spec(''K'') → ''X'', ''K'' some field, is equivalent to giving a point ''x'' ∈ ''X'' and an

extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * Ext ...

''K''/''k''(''x''). * The

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...

of a scheme of finite type over a field is equal to the transcendence degree of the residue field of the generic point.

Definition

Example

Properties

References

Further reading