Fandom

Scratchpad

II.4.8

216,193pages on
this wiki
Add New Page
Discuss this page0 Share

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.

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

Also on Fandom

Random wikia