- Residue field
In
mathematics , the residue field is a basic construction incommutative algebra . If "R" is acommutative ring and "m" is amaximal ideal , then the residue field is thequotient ring "k" = "R"/"m", which is a field. Frequently, "R" is alocal ring and "m" is then its unique maximal ideal.This construction is 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 abstractalgebraic variety is the 'natural domain' for the coordinates of the point.Definition
Suppose that "R" is a commutative
local ring , with the maximal ideal "m". Then the residue field is the quotient ring "R"/"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 "U" = Spec "A", with "A" some
commutative ring . Considered in the neighbourhood "U", the point "x" corresponds to aprime ideal "p" ⊂ "A" (seeZariski topology ). The "local ring " of "X" in "x" is by definition thelocalization "R" = "Ap", with the maximal ideal "m" = "p·Ap". 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". [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.]
A point is called "K"-rational for a certain field "K", if "k"("x") ⊂ "K".
Example
Consider the
affine line "A1k = Spec k [t] " over a field "k". If "k" is algebraically closed, there are exactly two types of prime ideals, namely*("t" − "a"), a ∈ 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" = ℝ, then the prime ideal ("x"2 + 1) has residue field isomorphic to ℂ.
Properties
* For a scheme 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 thegeneric point , havingtranscendence degree 1 over "k".
* A morphism "Spec K → X", "K" some field, is equivalent to giving a point "x" ∈ "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.Notes
References
* | year=1977, section II.2
Wikimedia Foundation. 2010.