This problem has H = 1, V =1
- part a), locally it is determined by so not only is it of finite type, but it is also finite.
- part b), locally its and the latter is a fintely generated A-algebra, and quasi compactness means can assume finite, etc.
- part c), basically reduces to showing if is a finitely generated algebra, and is a finitely generated algebra then is a fint. gen. A algebra.
part d) Say is of finite type and is any morphism.
- For that is covered by finitely many algebras in show for a suitable affine that the preimage in is covered by
- Show these algebras are finitely generated over .
- part e) follows from c,d
- part f) follow your nose and use II.3.3a,c
- part g) the point here is that a finitely generated algebra over a noetherian ring is also noetherian.