Ok, I feel a little bit ashamed by my question.


This afternoon in the train, I looked for a counter-example:<br/>
— k a field <br/>
— A a finitely generated k-algebra <br/>
— B a sub-k-algebra of A that is not finitely generated <br/>

Finally, I have found this: <br/>
— k any field<br/>
— A=k[x,y]<br/>
— B=k[xy, x.y^2, x.y^3, ...]<br/>

(proof : exercise)

My questions are : 

1) What is your usual counter-example ?<br/>
2) Under which conditions can we conclude that B is f.g. ?<br/>
3) How would you interpret geometrically this counter-example ?