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II.4.2

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This has H = 2; II.3.13d can be used to solve it.

HAPPY

Let U\subset X be the open set where the two morphisms agree; let  i \colon U \to Y be the composition.

  • From the assumption construct a map  h \colon X \to Y \times Y; let h' be its restriction to U
  • Argue (perhaps by a universal property) that h' = \Delta \circ i, in particular there is a factorization  X \to \Delta(Y) \to Y\times Y
  • use that X is reduced and II.3.11d to show in fact  X \to \overline{h(X)} \to \Delta(Y) \to Y \times Y
  • apply either projection to get a morphism h'' \colon X \to Y; it follows  h = \Delta \circ h''
  • play around with stuff like  f = p_1 \circ h to show  f = g
  • One example to show all assumptions are necessary:  X = Y = k[x,y]/(x^2, xy). Consider the maps  X \to Y given by the ring map in the other direction: x \mapsto 0, y \mapsto y and  x \mapsto x, y \mapsto y, find an open set where they agree. For an exmaple with  Y not separated let it be the affine line with the doubled origin and  X affine line.

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