II.4.8

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 $$ X \times Z$$

$$\begin{array}{ccc} X & \xrightarrow{f} & Y \\ \downarrow & & \downarrow \\ X \times_Z Y & \to & Y \times Y \end{array}$$ where the first vertical map is the graph of f and the second vertical map is the diagonal (which is a closed immersion).
 * for part e) consider the diagram
 * 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