Suppose my space $X$ is a K3 surface, hence has trivial canonical bundle. Given two coherent sheaves $F, E$ on $X$, we may define $\chi(E,F) = \sum_i (-1)^i \mathrm{ext}^i(E,F)$. I've read without proof that $\chi(E,F) = \chi(E^* \otimes F)$. I was wondering if someone could explain where this comes from?
I thought it must come from Serre duality, but then when I calculate I get the following: $\chi(E,F) = \sum_i (-1)^i \mathrm{ext}^i(E,F) = \sum_i (-1)^i \mathrm{ext}^i(E \otimes F^*, O_X) = \sum_i (-1)^i h^{2-i}(X,E \otimes F^*) = \sum_j (-1)^j h^j(X,E\otimes F^*) = \chi(E \otimes F^*)$.