Dealing with $\mathrm{Ext}^i(\mathcal{F},k(x))$ on a smooth variety over a field $k$, with $\mathcal{F}$ coherent and $k(x)$ skyscraper sheaf of a closed point I foundin a proof that for $i=2,3$ (and hence i guess for $i \geq 2$) it vanishes. I know that it reduces to a question about $\mathrm{Ext}^i(M,A/\mathfrak{m})$ where $M$ is a finitely generated module (i.e. the stalk at $x$ of $\mathcal{F}$) and $(A,\mathfrak{m})$ is a noetherian regular local ring.
How can I prove what I found claimed? Also, do I need all the hypothesis I have or something less is enough? I was not sure whether to approach it directly, or whether to argue something from some exact sequence, such as $0 \rightarrow \mathfrak{m} \rightarrow A \rightarrow A/\mathfrak{m} \rightarrow 0$ or a locally free resoulution of $M$, but after scribbling something on a piece of paper I did not get far with these ideas...