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  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

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