Is every open affine subscheme of an algebraic $k-$variety an affine $k-$variety?

Question

Note that an affine $k-$variety is a scheme isomorphic to the spectrum of a finitely generated $k-$algebra. So I am just asking the following: if $U=\text{Spec} A$ is an affine open subset of an algebraic $k-$variety, and $A$ is a $k-$algebra, is it finitely generated?

Answer 1

In general, if $f\colon X\to Y$ is a morphism (locally) of finite type, then for any open affine $\operatorname{Spec}(B)$ of $X$ mapping into an open affine $\operatorname{Spec}(A)$ of $Y$, the corresponding ring morphism $A\to B$ is of finite type.

Now, an algebraic $k$-variety is by definition a finite type $k$-scheme, i.e. the structure morphism $X\to \operatorname{Spec}(k)$ is of finite type. Now given $U=\operatorname{Spec}(A) \subset X$, it clearly maps into $\operatorname{Spec}(k)$ and hence the corresponding morphism $k\to A$ is of finite type, i.e. $A$ is a finitely generated $k$-algebra.