Relation between global Ext for sheaves on $\mathbb{P}^n$ and Ext for graded modules Let $S = \mathbb{C}[z_0,\dots,z_n]$, let $F$ be a finitely generated graded $S$-module, and let $\mathcal{F}$ be its associated coherent $\mathcal{O}_{\mathbb{P}^n}$-module.
Question:
What is the relation between $\text{Ext}_S^k(F,S)$ and global Ext of the sheaves $\mathcal{F}$ and $\mathcal{O}_{\mathbb{P}^n}$: $\text{Ext}^k(\mathcal{F},\mathcal{O}_{\mathbb{P}^n})$?
I am confused by the following. By local duality, and Serre duality, it seems that
$$
\text{Ext}_S^k(F,S)_d \cong (H_\mathfrak{m}^{n+1-k}(F)_d)^* \cong (H^{n-k}(\mathbb{P}^n,\mathcal{F}(d)))^* \cong \text{Ext}^k(\mathcal{F}(d),\mathcal{O}_{\mathbb{P}^n})
$$
for $0 \leq k < n$.
This seems strange since, if I understand correctly, $\text{Ext}^k(\mathcal{F}(d),\mathcal{O}_{\mathbb{P}^n})$ cannot be computed starting with a locally free resolution of $\mathcal{F}(d)$. However, $\text{Ext}_S^k(F,S)$ can be computed starting from a graded free resolution of $F$:
$$
0 \to F_n \to \dots \to F_0 \to F.
$$
Since the maps are matrices of polynomials this corresponds to a locally free resolution of $\mathcal{F}(d)$:
$$
0 \to \widetilde{F}_n(d) \to \dots \to \widetilde{F}_0(d) \to \mathcal{F}(d).
$$
 A: This is too long for a comment. Take the first step of a resolution $0\to G\to F\to \mathcal{F}\to 0$, where $F$ is a direct sum of line bundles. Assume $2\leq k<n$. Then, the long exact sequence gives, 
$$\operatorname{Ext}^{k-1}(F,\mathcal{O})\to\operatorname{Ext}^{k-1}(G,\mathcal{O})\to\operatorname{Ext}^k(\mathcal{F},\mathcal{O})\to\operatorname{Ext}^k(F,\mathcal{O}).$$ Notice that the first and last term are zero since $\operatorname{Ext}^i(F,\mathcal{O})=H^i(F^*)=0$ for $0<i<n$. Thus the problem reduces to $k=0,1$ which I will leave you to check can be theoretically understood in terms of the first step of a resolution.
A: Whenever $\mathcal F = \tilde M$ is a coherent sheaf given by a graded module $M$ over the ring $S = \mathbb C[x_0,\dots,x_n]$, it is possible to compute the cohomology of $\mathcal F$ from a graded free resolution of $M$. Only, that this is more complicated. The recipe can probably be extracted from Hartshorne's "Residues and Duality". There is a more down-to-earth approach sketched e.g. in the last section of this paper which makes this a little more explicit in terms of Cech cohomology.
