Affine algebraic sets and reduced affine k-algebras at the object level

Let $X\subseteq {\mathbf{A}}_k^n$ be an affine algebraic set. By [L1], the ring $k[X]$ is a quotient of the polynomial ring $k[x_1,\dots ,x_n]$, so [L3] shows that $k[X]$ is of finite type over $k$.

Let $A$ be a reduced affine $k$-algebra. By [L2] and [L3], there exist $n\ge 0$ and an ideal $I\subseteq k[x_1,\dots ,x_n]$ with $A\cong k[x_1,\dots ,x_n]\mathrm{/}I$. Because $A$ is reduced, if $f^m\in I$ then the class of $f$ in $A$ is nilpotent, hence zero; therefore $f\in I$. So $I$ is a radical ideal.

To see that $k[X]$ is reduced, let $\bar{f}\in k[X]$ satisfy ${\bar{f}}^m=0$ for some $m\ge 1$. Choosing a representative $f$ in the polynomial ring, this means $f^m\in I(X)$. Since $I(X)$ is radical by [L4], we have $f\in I(X)$, so $\bar{f}=0$. Thus $k[X]$ is reduced. Together with step 1.1 and [L2], this proves part (1).

By [L4], the radical ideal $I$ is the vanishing ideal of the algebraic set $X:=V(I)\subseteq {\mathbf{A}}_k^n$. Then [L1] gives $k[X]=k[x_1,\dots ,x_n]\mathrm{/}I(X)=k[x_1,\dots ,x_n]\mathrm{/}I$, so $A\cong k[X]$.

Steps 2.1 and 2.2 establish the stated object-level correspondence between affine algebraic sets and reduced affine $k$-algebras.