A reference to the Riemann-Roch theorem can be found here: http://en.wikipedia.org/wiki/Riemann-Roch and here: http://en.wikipedia.org/wiki/Grothendieck%E2%80%93Riemann%E2%80%93Roch_theorem Would you know any reference that introduces Grothendieck-Riemann-Roch for $Ext$? (If it includes a reference for $Ext$ of elements in a derived category, all the better. But, if not, then that's also pretty good.)

I am looking for a reference of this formula because in my current work I would like to apply the following argument. I have a bunch of $Ext$ groups: $Ext^0(I^\bullet,F), Ext^1(I^\bullet,F), Ext^2(I^\bullet,F)$, where $F$ is a sheaf over a surface $X$ and $I^\bullet$ is a chain complex of sheaves over $X$ (if that is confusing, just think $I^\bullet$ is a sheaf.) In my case, one can show that $Ext^2(I^\bullet, F)$ vanishes and that the dimension of $Ext^0(I^\bullet,F)$ is constant. I would like to conclude that $Ext^1(I^\bullet,F)$ also has constant dimension.

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    $\begingroup$ Perhaps, would Theorem 14 in arXiv:0707.2052 [Caldararu-Willerton] be good for you? $\endgroup$
    – S. Okada
    Jun 18, 2010 at 14:41

1 Answer 1


One has $Ext^\bullet(I,F) = H^\bullet(I^*\otimes F)$, where the dualization and the tensor product are derived and $H^\bullet$ stands for the hypercohomology. The Euler characteristic can be computed via Riemann--Roch, so if you know that all $Ext$'s with exception of two vanish, and one of this two has constant dimension, then you can conclude that the other one has constant dimension as well.

  • $\begingroup$ Dear Sasha, thanks very much indeed for your help. Would you, by any change, know a nice reference for dualization on derived category? Thanks a lot. $\endgroup$
    – James O
    Jun 20, 2010 at 22:02

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