II.3.11

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.