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?)