It is known (although maybe not so well) that there are nice algebraic varieties whose ring of global functions is not finitely generated over the ground field. One can find examples on the web, but they are somewhat scarce. In thinking about this question, one can try to describe the ring of functions as an algebra over the function ring of another variety. Eventually this lead me to the following question:
Let $S$ be an affine noetherian scheme and let $U\subset S$ be an open subscheme. When is the map of function rings $f:\mathcal{O}(S)\to\mathcal{O}(U)$ finitely generated? flat? étale?
Comments:
1) example: the map $f$ is étale if $S$ is integral, S2 and every divisor is locally set-theoretically principal. Indeed, in this case by the S2 condition only the codimension 1 part of $S\setminus U$ contributes to the function ring, and then locally $U$ is a principal open. This includes locally factorial schemes, reduced curves, and the affine quadric cone $xy=z^2$.
2) I'm interested also in reducible and/or nonreduced examples. I'm interested in all three properties (f.g., flat, étale). All kinds of examples and counterexamples are very welcome.
3) The similar MO question MO329902 did not receive answers...
