# 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.