Is infiniteness of $K$ essential for the isomorphism between coordinate rings and polynomial functions?

Question

Let $S$ be an algebraic sets in n-dimensional affine space, $K$ an infinite field, an exercise in Aluffi’s Algebra:Chapter 0 (page 414, ex 2.12) asks a proof that there is an isomorphism of $K$- algebra between the polynomial functions on $S$ and the coordinate rings of $S$.

I can’t understand the necessity of infiniteness of $K$, why without it the identification will not be available ?

Answer 1

A very simple counterexample: on the prime field of characteristic $p$, the Frobenius map $\;\mathbf F_p\longrightarrow\mathbf F_p$, $x\longmapsto x^p$ is equal to the identity. Yet, in the polynomial ring $\mathbf F_p[X]$, $X^p\ne X$.

Answer 2

Let's say $K = \mathbb{F}_p$ and $S$ is just $\mathbb{A}^1$. Then the function $x \mapsto x^p$ is the identity on $S$ but it's not the case that $x^p$ is equal to $1$ in the coordinate ring $k[x]$.