词条 | Residue field |
释义 |
In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field.[1] Frequently, R is a local 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).[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.{{clarify|date=February 2015}} DefinitionSuppose 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 a prime ideal p ⊂ A (see Zariski topology). The local ring of X in x is by definition the localization R = Ap, with the maximal ideal m = p·Ap. Applying the construction above, we obtain the residue field of the point x : k(x) := Ap / p·Ap. One can prove that this definition does not depend on the choice of the affine neighbourhood U.[3] A point is called K-rational for a certain field K, if k(x) ⊂ K.[4] ExampleConsider the affine line A1(k) = Spec(k[t]) over a field k. If k is algebraically closed, there are exactly two types of prime ideals, namely
The residue fields are
If k is not algebraically closed, then more types arise, for example if k = R, then the prime ideal (x2 + 1) has residue field isomorphic to C. Properties
References1. ^{{cite book| last1 = Dummit| first1 = D. S.| last2 = Foote| first2 = R.| title = Abstract Algebra| publisher = Wiley| year = 2004| edition = 3| isbn = 9780471433347 }} 2. ^{{cite book | author = David Mumford | year = 1999 | title = The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians | edition = 2nd | publisher = 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. ^Görtz, Ulrich and Wedhorn, Torsten. Algebraic Geometry: Part 1: Schemes (2010) Vieweg+Teubner Verlag. Further reading
1 : Algebraic geometry |
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