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.