The 'usual' Hirzebruch-Riemann-Roch theorem gives a topological expression for $\chi(E)$, where $E$ is a coherent sheaf on a smooth projective variety. Is there a generalisation of this giving a topological expression for $$\chi(E,F)=\sum\_i (-1)^i \mathrm{Ext}^i(E,F)?$$
By definition $\chi(E)=\chi(\mathcal{O},E)$, and if $E$ is locally free, then of course $\chi(E,F)=\chi(\mathcal{Hom}(E,F))$, but in my case of interest $E$ is not locally free.



Generalized Riemann Roch theorem
If the variety $X$ is smooth, one has
$$
\chi(E,F) = \deg\Big( \mathrm{ch}(E)^\vee \cdot \mathrm{ch}(F) \cdot \mathrm{td}\_X \Big),
$$
where $(-)^\vee$ is the involution of $\bigoplus H^{2i}(X,\mathbb{Q})$ that acts by $(-1)^i$ on the summand $H^{2i}(X,\mathbb{Q})$.


