# II.3.13

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This problem has H = 1, V =1

### HAPPY

• 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.

part d) Say $X \to Y$ is of finite type and $Y' \to Y$ is any morphism.

• 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 e) follows from c,d