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 ?



Euler characteristic and inclusion-exclusion
$$
0\to \mathscr O\_{C\_1\cup C\_2} \to \mathscr O\_{C\_1}\oplus \mathscr O\_{C\_2} \to \mathscr O\_{C\_1\cap C\_2} \to 0
$$
with maps $a\mapsto (a,a)$ and $(a,b)\mapsto a-b$  

is exact and $\chi$ is additive.


