coordinate ring

Last updated January 27, 2022

Definition

Let $V$ be an Algebraic subset of $k^n$, $k$ algebraically closed.

• The coordinate ring of $V$, denoted $k[V]$, is the $k$-algebra of $k$-valued functions on $V$ obtained by restricting polynomial functions to $V$.

Properties

1. Letting $I(V)$ be the ideal of functions which vanish on $V$, $$k[V]\cong k /I(V).$$
2. As sets, \begin{gather}\text{ev}:V\overset{\simeq}{\rightarrow}\text{Hom}_{k-\text{alg}}(k[V],k),\ x\mapsto \text{ev}_x \ (x_i=\phi(X_i|_V))\leftarrow \phi\end{gather} where the hom-set consists of $k$-algebra homomorphisms.
   1. Let $\text{Specm}(A)$ denote the set of maximal ideals of $A$. By Nullstellensatz, the above says \begin{gather}\text{Hom}_{k-\text{alg}}(k[V],k)\cong\text{Specm}(k[V]), \ f\mapsto\text{ker}(f).\end{gather} (Indeed, $\text{Specm}(k[V])\cong V$.)
3. $k[V]$ is finitely generated and reduced (i.e. does not contain non-zero nilpotent elements).

Proofs 2