Suppose $V$ is a quasi-affine variety over an algebraically closed field $k$. I read that the algebra $k[V]$ is isomorphic to a subalgebra of some finitely generated $k$-algebra.
However, I thought $k[V]$ is already a finitely generated $k$-algebra, making the claim trivial. I suspect there's something more to it, else it wouldn't be worth mentioning. If there is more meat to this claim, does anyone have a proof or reference to a proof? Thanks.
It's not obvious that $k[V]$ is finitely generated. If $V$ is closed then it's just $k[\mathbb{A}^n]/I(V)$, but otherwise we define $k[V]$ as the functions given locally by well-defined elements of $k(\mathbb{A}^n)$, which is not finitely generated. In fact there are examples of quasi-projective varieties with non-finitely generated rings of functions, for instance, the total space of the sum of a degree $0$ nontorsion and a positive-degree line bundle on an elliptic curve. You can find this example described in notes of Ravi Vakil, who also varies it to give a quasi-affine example.
It's much easier to show that a quasi-affine algebra is contained in a finitely generated algebra than to construct an example as discussed above. Take a cover of a quasi-affine variety $X$ by finitely many open affine subsets $U_i$, yielding an epimorphism $\sqcup U_i\to X$ and thus a monomorphism $k[X]\to k[\sqcup U_i]\cong \prod k[U_i]$.