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This has H = 2, V = ... (II.2.3 can be used to solve this)
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 both have P; set
- apply base change to and to to get having P
- similarly get having P then apply base change with these morphisms to get a unique map ( which is a composition of things having P) (this all looks better when written out as diagrams)
- The thing to show is that , one way to do this is just show has the universal property of
- for part e) consider the diagram
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. is the fiber product of the bottom map and the right vertical map)
- use has P to conclude that is obtained by base change hence has P, conclude that f has P
- for part f), prove a lemma that is a closed immersion (hence has P)
- show has P (composition of things in P)
- show this is the same as hence this also has P
- use e) to get the result