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Coordinate Ring

Given an affine variety V in the n-dimensional affine space K^n, where K is an algebraically closed field, the coordinate ring of V is the quotient ring

K[V]=K[x_1,...,x_n]/I(V),

where I(V) is the ideal formed by all polynomials f(x_1,...,x_n) with coefficients in K which are zero at all points of V. If V is the entire n-dimensional affine space K^n, then this ideal is the zero ideal. It follows that the coordinate ring of K^n is the polynomial ring K[x_1,...,x_n]. The coordinate ring of a plane curve defined by the Cartesian equation f(x_1,x_2)=0 in the affine plane K^2 is K[x_1,x_2]/<f(x_1,x_2)>.

In general, the Krull dimension of ring K[V] is equal to the dimension of V as a closed set of the Zariski topology of K^n.

Two polynomials f(x_1,...,x_n) and g(x_1,...,x_n) define the same function on V iff f-g in I(V). Hence the elements of K[V] are equivalence classes which can be identified with the polynomial functions from V to K.

This entry contributed by Margherita Barile

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References

Shafarevich, I. R. Basic Algebraic Geometry 1 and 2, 2nd ed. Berlin: Springer-Verlag, 1994.

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Coordinate Ring

Cite this as:

Barile, Margherita. "Coordinate Ring." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CoordinateRing.html

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