II.4.8

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This has H = 2, V = ... (II.2.3 can be used to solve this)

HAPPY

The set of morphisms having P are such that closed immersions have P, P is closed under composition, and morphisms in P are stable under base change

• for part d, say $X\to Y, Z \to W$ both have P; set $T = Y \times W$
  • apply base change to $X \to Y$ and to $T \to Y$ to get $X \times_Y T \to T$ having P
  • similarly get $Z \times_W T \to T$ having P then apply base change with these morphisms to get a unique map ( which is a composition of things having P) $L := (X \times_Y T)\times_T(Z \times_W T)$ (this all looks better when written out as diagrams)
• The thing to show is that $L\cong X \times Z$, one way to do this is just show $L$ has the universal property of $X \times Z$

• for part e) consider the diagram

$\begin{array}{ccc} X & \xrightarrow{f} & Y \\ \downarrow & & \downarrow \\ X \times_Z Y & \to & Y \times Y \end{array}$

where the first vertical map is the graph of f and the second vertical map is the diagonal (which is a closed immersion).

• show this diagram is cartesian (i.e. $X$ is the fiber product of the bottom map and the right vertical map)
• use $f \circ g$ has P to conclude that $X \times_Z Y \to Y$ is obtained by base change hence has P, conclude that f has P

• for part f), prove a lemma that $X_{red} \to X$ is a closed immersion (hence has P)
  • show $X_{red} \to X \to Y$ has P (composition of things in P)
  • show this is the same as $X_{red} \to X \to Y_{red} \to Y$ hence this also has P
  • use e) to get the result