An affine scheme of finite type over a field has a finitely generated coordinate ring

By [F1] the morphism $X\to \mathrm{Spec}k$ is quasi-compact and locally of finite type, so its affine open charts $\mathrm{Spec}B$ with $B$ a finitely generated $k$-algebra cover $X$; quasi-compactness extracts finitely many of them, giving $X={\bigcup }_{i=1}^n{U}_i$ with $U_i=\mathrm{Spec}{B}_i$ and $B_i$ finitely generated over $k$.

Each $U_i$ is affine, hence quasi-compact by [F4], and by [F2] every point of $U_i$ lies in some distinguished open $D(f)\subseteq U_i$ with $f\in A$; finitely many such sets cover $U_i$. Collecting the finitely many results over $i=1,\dots ,n$, there are $f_1,\dots ,f_m\in A$ with $X={\bigcup }_{j=1}^{m}D(f_j)$ and $D(f_j)\subseteq U_{i(j)}$ for a suitable index $i(j)$.

Fix $j$ and put $i=i(j)$, so that $D(f_j)\subseteq U_i=\mathrm{Spec}{B}_i$; let $b\in B_i$ be the restriction of the global section $f_j$ to $U_i$. A prime $\mathfrak{q}\in U_i$ has image $\mathfrak{p}=\mathfrak{q}\cap A$ under the open immersion, so $f_j\mathrm{\notin }\mathfrak{p}$ if and only if $b\mathrm{\notin }\mathfrak{q}$, and therefore $D(f_j)=D(b)$ as open subsets of $U_i$. The structure sheaf of $X$ restricted to $U_i$ is that of $U_i$, so by [F3] $A_{f_j}=\mathrm{\Gamma }(D(f_j),{\mathcal{O}}_X)=\mathrm{\Gamma }(D(b),{\mathcal{O}}_{U_i})=(B_i{)}_b$, a principal localization of a finitely generated $k$-algebra and hence again a finitely generated $k$-algebra.

For each $j$ choose finitely many $k$-algebra generators of $A_{f_j}$ and write them as fractions with numerators in $A$. By [F5] choose a unit relation $1={\sum }_j{\lambda }_j{f}_j$ with ${\lambda }_j\in A$. Let $B\subseteq A$ be the $k$-subalgebra generated by all $f_j$, all these numerators, and all ${\lambda }_j$. It is finitely generated, and the natural map $B_{f_j}\to A_{f_j}$ is surjective for each $j$, since its image contains the chosen algebra generators.

Fix $a\in A$. For every $j$, surjectivity in step 4.1 gives $a\mathrm{/}1=b_j\mathrm{/}{f}_j^{r_j}$ in $A_{f_j}$ for some $b_j\in B$; equality of localization fractions gives $f_j^{t_j}(f_j^{r_j}a-b_j)=0$ in $A$ for some $t_j\ge 0$. Thus $f_j^{r_j+t_j}a=f_j^{t_j}{b}_j\in B$. Choose $N\ge 1$ exceeding the finitely many exponents $r_j+t_j$; then $f_j^{N}a\in B$ for all $j$. Expanding $({\sum }_j{\lambda }_j{f}_j{)}^{mN}=1$ shows $1={\sum }_j{c}_j{f}_j^N$ with $c_j\in B$, because each monomial has some $f_j$-exponent at least $N$ and all ${\lambda }_j,f_j$ belong to $B$. Hence $a={\sum }_j{c}_j(f_j^{N}a)\in B$. This argument chooses $N$ separately for each $a$, and proves $A=B$. If $X$ is empty, the same unit-ideal conclusion gives $A=0$, which is finitely generated as well.

For the consequence, if $G$ is a group scheme of finite type over $k$ whose underlying scheme is affine, say $G=\mathrm{Spec}A$, then by definition its structure morphism $G\to \mathrm{Spec}k$ is of finite type, so the argument above makes its coordinate ring finitely generated over $k$. The AC-dependent inputs were affine-chart quasi-compactness [F4] and the unit-ideal relation [F5]; the subsequent selections were finite.