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Let $X$ be a smooth projective surface over $\mathbb{C}$. How to show that for any zero-dimensional subscheme $Z$ of $X$, the Euler characteristic $\chi(\mathcal{O}_Z,\mathcal{O}_Z)={\sum}_i (-1)^i \ dim \ Ext^i_{\mathcal{O}_X} (\mathcal{O}_Z,\mathcal{O}_Z)$ is the same for all $Z$ of length $n$?

(I read that one could use a form of the Hirzebruch-Riemann-Roch theorem $\chi(\mathcal{O}_Z,\mathcal{O}_Z)= {\int}_X ch(\mathcal{O}_Z) ch(\mathcal{O}_Z)^{\vee} td(X)$ to compute this Euler characteristic, but I am wondering if there is a more elementary way to see this? If not, is there a reference for this form of Hirzebruch-Riemann-Roch?)

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up vote 6 down vote accepted

Define $\chi(\mathcal M,\mathcal N)$ for all finite length $\mathcal O_X$-modules. It is additive in both arguments so for its computation we get $$ \chi(\mathcal M,\mathcal N) = \sum_{x,y} \mathrm{lgth}_x(\mathcal M)\mathrm{lgth}_y(\mathcal N)\chi(k(x),k(y)). $$ Clearly $\chi(k(x),k(y))=0$ if $x\neq y$ and the exactness of the Koszul resolution shows that $$ \chi(k(x),k(x)) = \sum_i(-1)^i\binom{n}{i} = (1-1)^n=0, $$ where we let $X$ be smooth and $n$-dimensional. Hence we actually get $\chi(\mathcal M,\mathcal N)=0$ for all $\mathcal M$ and $\mathcal N$ of finite length.

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