Define the Euler characteristic of a scheme to be the Euler characteristic of its structure sheaf. I remember being told that for curves, this invariant satisfies inclusion-exclusion. That is, if $C_1, C_2$ are curves , then

$$\chi(C_1 \cup C_2) + \chi(C_1\cap C_2) = \chi(C_1) + \chi(C_2)$$

The intersection is scheme theoretic.Does any one knows a proof or a reference for this ?


1 Answer 1


$$ 0\to \mathscr O_{C_1\cup C_2} \to \mathscr O_{C_1}\oplus \mathscr O_{C_2} \to \mathscr O_{C_1\cap C_2} \to 0 $$ with maps $a\mapsto (a,a)$ and $(a,b)\mapsto a-b$
is exact and $\chi$ is additive.

  • $\begingroup$ Why is the latter map surjective? $\endgroup$ Dec 12, 2011 at 9:30
  • 2
    $\begingroup$ It's surjective individually already. $C_1\cap C_2$ is a closed subscheme of $C_i$ for $i=1,2$, so $\mathscr O_{C_i}\to \mathscr O_{C_1\cap C_2}$ is surjective. So if $a$ in $\mathscr O_{C_1\cap C_2}$ lifts to $a$ in $\mathscr O_{C_1}$, then it is the image of $(a,0)$ by the above map. $\endgroup$ Dec 12, 2011 at 10:17
  • $\begingroup$ Oh yes, sorry for the stupid question. The left exactness is equivalent to the generalized Chinese Remainder Theorem which I have mentioned here mathoverflow.net/questions/10014. So in fact $C_1,C_2$ don't have to be curves, but they can be arbitrary closed subschemes. $\endgroup$ Dec 12, 2011 at 11:26
  • $\begingroup$ Yes, right. I meant to mention that it works in any dimension (and they don't have to be equi-dimensional either). $\endgroup$ Dec 12, 2011 at 11:31

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