Calculating Hom in derived category

Question

I got stuck calculating $Hom^* (\mathcal O, \mathcal O(k)) \in D(Coh(\mathbb P^n))$. On one hand, $Ext^i (\mathcal O, \mathcal O(k)) = H^i (\mathcal O^* \otimes \mathcal O(k)) = H^i (\mathcal O(k))$, so there is $Ext^n$ if $k << 0$. On other hand, there is projective resolution $0 \to \mathcal O \to \mathcal O \to$, which gives $Ext^{>0} = 0$. Where am I wrong and how to calculate (i.e. show a complex in the class of) $Hom^*$ in derived category?

Answer 1

In the non-affine case (see Serre's criterion for affine-ness), the structure sheaf ${\mathscr O}$ is not projective: You have $\text{Hom}({\mathscr O},-)=\Gamma(X;-)$, which is not exact, and whose failure of exactness is measured by $\text{H}^*(X;-)$. What's more, there isn't even any nontrivial projective object in the category of coherent sheaves over ${\mathbb P}^n$, see e.g. Representations of a quiver and sheaves on P^1.