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II.3.11

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This exercise has H = 2; ex II.2.4 can be used; V = ....


HAPPY

For part a, let f \colon X \to Y be a closed immersion and  g\colon Y' \to Y any morphism.

  • Either argue locally and glue, or show Y'\times X \cong g^{-1}(g(Y') \cap f(X)) the latter be a closed subset of Y'
    • Arguing locally would involve showing that if  f is locally A \to A/I and g is locally \cup_i \mbox{Spec } A'_i \to \mbox{Spec } A, then the fiber product is locally \cup_i \mbox{Spec } A'_i \otimes_A A/I \cong \cup_i \mbox{Spec } A'_i/I.
  • I'd say skip b) in favor of a much simpler proof using qcoh sheafs of ideals presented in section II.5.
  • part c, the statement of topological spaces is clear, it remains to show the statement on the level of sheaves, reduce to the affine case:  A \to A/I^n \to A/I
  • part d, to determine what  Y should be just look locally.
    • Let U = \mbox{Spec } A \subset X, then f^{-1}(U) = V is an open set and the morphism here is determined by A \to \Gamma(V, O_V) (ex. II.2.4). Take the kernel to construct a sheaf of ideals in X that will define Y.

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