II.3.13

This problem has H = 1, V =1

HAPPY
part d) Say $$ X \to Y$$ is of finite type and $$ Y' \to Y$$ is any morphism.
 * part a), locally it is determined by $$ A \to A/I$$ so not only is it of finite type, but it is also finite.
 * part b), locally its $$A \to A_a$$ and the latter is a fintely generated A-algebra, and quasi compactness means can assume finite, etc.
 * part c), basically reduces to showing if $$ B $$ is a finitely generated $$A$$ algebra, and $$C$$ is a finitely generated $$B$$ algebra then $$C$$ is a fint. gen. A algebra.
 * For $$ \mbox{Spec A} \subset Y$$ that is covered by finitely many $$B_i$$ algebras in $$X$$ show for a suitable affine $$ \mbox{Spec A'} \subset Y'$$ that the preimage in $$ X\times_Y Y' $$ is covered by $$ A' \otimes_A B_i$$
 * Show these algebras are finitely generated over $$A'$$.
 * part d) follows from c,d
 * part e) the point here is that a finitely generated algebra over a noetherian ring is also noetherian.