Let $F,G$ be coherent sheaves on a smooth Noetherian scheme $X$ of dimension $n$. I can see that $Ext^k(F,G) = 0$ for $k > 2n$. Is this sharp? Or can it perhaps be improved to get vanishing for $k>n$?
The argument I have in mind is to use the spectral sequence $H^i(\mathcal{Ext}^j(F,G)) \Rightarrow Ext^{i+j}(F,G)$. Since $\mathcal{Ext}^j(F,G)$ can be computed locally on affine charts using a projective resolution, we have $\mathcal{Ext}^j(F,G) = 0$ for $j > n$ by smoothness. Moreover $\mathcal{Ext}^j(F,G)$ is coherent, so by the Grothendieck vanishing theorem we have $H^i(\mathcal{Ext}^j(F,G)) = 0$ for $i > n$. So the $E_2$ page of the spectral sequence is already zero outside the box $(i,j) \in [0,n] \times [0,n]$, and in particular for $i+j > 2n$.
But perhaps something more subtle happens?