# Generalized Riemann Roch theorem

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.

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 $$H^{2i}(X,\mathbb{Q})$$.
• Thanks! I guess the proof should be similar to the one for the 'usual' Riemann-Roch theorem, using that $\chi(E,F)$ is additive in $E$ as well? Is there a reference for this? – Bernie Nov 13 '19 at 14:50