Residue field

In mathematics, the residue field is a basic construction in commutative algebra. If ${\displaystyle R}$ is a commutative ring and ${\displaystyle {\mathfrak {m}}}$ is a maximal ideal, then the residue field is the quotient ring ${\displaystyle k=R/{\mathfrak {m}}}$, which is a field. ^[1] Frequently, ${\displaystyle R}$ is a local ring and ${\displaystyle {\mathfrak {m}}}$ 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 ${\displaystyle x}$ of a scheme ${\displaystyle X}$ one associates its residue field ${\displaystyle k(x)}$. ^[2] 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.^{[ clarification needed ]}

Definition

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

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

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

One can prove that this definition does not depend on the choice of the affine neighbourhood ${\displaystyle {\mathcal {U}}}$. ^[3] Since localization is exact, ${\displaystyle k(x)}$ is the field of fractions of ${\displaystyle A/{\mathfrak {p}}}$ (which is an integral domain as ${\displaystyle {\mathfrak {p}}}$ is a prime ideal). ^[4]

A point is called ${\displaystyle \color {blue}k}$-rational for a certain field ${\displaystyle k}$, if ${\displaystyle k(x)=k}$. ^[5]

Example

Consider the affine line ${\displaystyle \mathbb {A} ^{1}(k)=\operatorname {Spec} (k[t])}$ over a field ${\displaystyle k}$. If ${\displaystyle k}$ is algebraically closed, there are exactly two types of prime ideals, namely

• ${\displaystyle (t-a),\,a\in k}$
• ${\displaystyle (0)}$, the zero-ideal.

The residue fields are

• ${\displaystyle k[t]_{(t-a)}/(t-a)k[t]_{(t-a)}\cong k}$
• ${\displaystyle k[t]_{(0)}\cong k(t)}$, the function field over k in one variable.

If ${\displaystyle k}$ is not algebraically closed, then more types arise from the irreducible polynomials of degree greater than 1. For example if ${\displaystyle k=\mathbb {R} }$, then the prime ideals generated by quadratic irreducible polynomials (such as ${\displaystyle x^{2}+1}$) all have residue field isomorphic to ${\displaystyle \mathbb {C} }$.

Properties

• For a scheme locally of finite type over a field ${\displaystyle k}$, a point ${\displaystyle x}$ is closed if and only if ${\displaystyle k(x)}$ is a finite extension of the base field ${\displaystyle k}$. This is a geometric formulation of Hilbert's Nullstellensatz. In the above example, the points of the first kind are closed, having residue field ${\displaystyle k}$, whereas the second point is the generic point, having transcendence degree 1 over ${\displaystyle k}$.
• A morphism ${\displaystyle \operatorname {Spec} (K)\rightarrow X}$, ${\displaystyle K}$ some field, is equivalent to giving a point ${\displaystyle x\in X}$ and an extension ${\displaystyle K/k(x)}$.
• 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

1. Dummit, D. S.; Foote, R. (2004). Abstract Algebra (3 ed.). Wiley. ISBN   9780471433347.
2. David Mumford (1999). The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians. Lecture Notes in Mathematics. Vol. 1358 (2nd ed.). Springer-Verlag. doi:10.1007/b62130. ISBN   3-540-63293-X.
3. 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.
4. Matsumura, Hideyuki (1980). Commutative Algebra. Mathematics Lecture Note Series. Vol. 56 (2nd ed.). Reading, Massachusetts: The Benjamin/Cummings Publishing Company Inc. ISBN   0-8053-7026-9., 1.K
5. Görtz, Ulrich and Wedhorn, Torsten. Algebraic Geometry: Part 1: Schemes (2010) Vieweg+Teubner Verlag.

Further reading

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