Localization of a finite type algebra is a localization of an integral domain of finite type

Question

Let $k$ be a field. Algebras are assumed to be commutative, associative, unital. If an integral domain over $k$ is a localization of a finitely generated $k$-algebra at a multiplicative set, is it also a localization of an integral domain that is of finite type over $k$ at a multiplicative set?

Answer 1

Yes. Indeed, suppose $A$ is a finitely generated $k$-algebra and $S\subset A$ is a multiplicatively closed set such that the localization $A[S^{-1}]$ is a domain. Let $I\subset A$ be the kernel of the localization map $A\to A[S^{-1}]$. Then $I$ is finitely generated, and each element of $I$ is annihilated by some element of $S$. We can thus choose a finitely generated multiplicatively closed subset $S_0\subseteq S$ such that every element of $I$ becomes $0$ in $A[S_0^{-1}]$. This means that the localization map $A[S_0^{-1}]\to A[S^{-1}]$ is injective. Thus $A[S_0^{-1}]$ is a domain which has $A[S^{-1}]$ as a localization, and it is finitely generated as a $k$-algebra.