Ok, I feel a little bit ashamed by my question.
This afternoon in the train, I looked for a counter-example:
— k$k$ a field
— A$A$ a finitely generated k$k$-algebra
— B$B$ a sub-k$k$-algebrasubalgebra of A$A$ that is not finitely generated
Finally, I have found this:
— k$k$ any field
— A=k[x,y]$A=k[x,y]$
— B=k[xy, x.y^2, x.y^3, ...]$B=k[xy, xy^2, xy^3, \dots]$
(proof : exercise)
My questions are :
- What is your usual counter-example ?
- Under which conditions can we conclude that B$B$ is f.g. ?
- How would you interpret geometrically this counter-example ?
