Category Archives: Algebraic Geometry

The Coordinate Ring

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I’m in the process of picking up some algebraic geometry from this thesis on Goppa codes. There’s a terminology I came across called the coordinate ring of an affine variety \mathcal X.

Let \mathbb K be an algebraically closed field. The n-dimensional affine space, denoted \mathbb A^n is the space of n-tuples of \mathbb K. An ideal I \subseteq \mathbb K[x_1, x_2, \ldots, x_n] corresponds to an algebraic set defined as

V(I) := \{(a_1, \ldots, a_n): F(a_1, \ldots, a_n) = 0 \quad \forall F \in I\}

If I \subsetneq \mathbb K[x_1, \ldots, x_n] is a prime ideal, the algebraic set V(I) is called an affine variety.

Let \mathcal X = V(I) where I is as above. The integral domain \mathbb K[\mathcal X] := \mathbb K[x_1, \ldots, x_n]/I is called the coordinate ring of the affine variety \mathcal X. The function field, denoted by K(\mathcal X), is the field of fractions of \mathbb K[\mathcal X].

The idea that’s novel to me is that it appears a lot of algebraic geometry terminology comes from differential geometry. In differential geometry, we refer to coordinate charts x: U \to \mathbb R^n of manifolds and the individual components of x are called coordinates. The situation in algebraic geometry is not too different, i.e., on \mathbb A^n_K, the standard “coordinate chart” has coordinates given by the n variables. The term “coordinate function” apparently means functions that we can build out of the coordinates — similar to the ring of smooth real-valued functions on a manifold. The coordinate ring is the ring of all such functions.

For varieties V \subseteq \mathbb A^n_K, we can understand coordinate functions as restrictions of functions on \mathbb A^n_K to the variety V. To quote an MSE comment, in other words, as we define a variety in terms of polynomials if we want to look at features of a variety, we have to define those in terms of the polynomials that exist specifically in that variety. Hence, the motivation for finding the coordinate ring in the first place, and by coordinates, we mean “those particular coordinates which are in the variety”.