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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
  • 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.

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