# Ext modules of coherent sheaves and associated modules

Let $\mathcal{E}$ and $\mathcal{F}$ be two coherent sheaves on a polarized projective variety $(X,\mathcal{O}_X(1))$.

Denote by $E=\Gamma_*(\mathcal{E})=\oplus_{k\in\mathbb{Z}}\mathcal{E}(k)$, $F=\Gamma_*(\mathcal{F})$, and $R= \Gamma_\ast(\mathcal{O}_X)$.

Is it true that $$\oplus_{k\in\mathbb{Z}}Ext^i_{\mathcal{O}_X}(\mathcal{E}, \mathcal{F}(k))=Ext^i_R(E, F)?$$

If the equality does not hold in general, when and to what extent it will hold?

I assume you mean to ask whether: $$\oplus_{k\in\mathbb{Z}}Ext^i_{\mathcal{O}_X}(\mathcal{E}, \mathcal{F}(k))=Ext^i_R(E, F)?$$ For this question, the answer is no, set $\mathcal{E} = \mathcal{O}_X$. Choose $\mathcal{F}$ such that $H^i(X, \mathcal{F}) \neq 0$ for some $i > 0$. Then obviously $Ext^i_R(R, F) = 0$. However $Ext^i_{\mathcal{O}_X}(\mathcal{E}, \mathcal{F}(0)) = H^i(X, \mathcal{F}) \neq 0$.
• What if we assume that $H^i(X, \mathcal{F})=0$ for all $i>0$? May 30, 2012 at 6:00
• This example will work as long as some $H^i(X, \mathcal{F}(k)) \neq 0$. May 30, 2012 at 10:34
• Sorry. What I really mean is that $H^i(X, \mathcal{F}{k})=0$ for all $k$. Thank you for your answer. May 30, 2012 at 10:50
• If $i = \dim \mathcal{F}$, then $H^i(X, \mathcal{F}(k))$ generally will not be zero for any $k \ll 0$. $$\text{ }$$ Furthermore, if $\dim \mathcal{F} = 0$ (ie, a skyscraper sheaf), then the module $F$ is never finitely generated (unless it is zero). The reason is we have non-zero infinitely negative degrees. May 30, 2012 at 22:26
• What you can do is the following. Choose a locally free resolution of $\mathcal{E}$ made up of $\oplus \mathcal{O}_X(m)$. Now try to figure out if the $\Gamma_*$ of this resolution is still exact (at least in the low degrees you care about). If it happens to still be exact, then you are are more or less fine... May 30, 2012 at 22:28