This is a follow up to my question : Applications of $Ext^n$ in algebraic geometry
In the case of $\mathcal{O}_X$-Modules it is clear that $Ext^i(\mathcal{O}_X, \mathcal{F}) \cong H^i(X,\mathcal{F})$ so I was wondering if there is any particular case where you can compute $H^1(X,\mathcal{F})$ using the formalism of extensions of degree 1.
I am specially interested in the case $\mathcal{O}_{\mathbb{P}^n}(m)$ since I want to prove M.Noether's $AF+BG$ theorem.
Here is an argument for $n=2$, the general case is not different. I will just write $\mathcal{O}$ instead of $\mathcal{O}_{\mathbb{P}^2}$. Let $0\to\mathcal{O}(m)\to E\to\mathcal{O}\to 0$ be an extension. Let $S=\oplus_k\mathrm{H}^0(\mathcal{O}(k))$ be the homogeneous coordinate ring. Applying this to our exact sequence, we get, $0\to S\to M\to S$, an exact sequence of graded $S$-modules, where $M=\oplus_k\mathrm{H}^0(E(k))$. The cokernel of $M\to S$ is supported only at the irrelevant maximal ideal and so, if not zero, we see that the image $I$ has depth exactly one and we have an exact sequence $0\to S\to M\to I\to 0$. Since $M$ has depth at least two, this implies depth of $S$ is two, which is false (this is where we use $n>1$). Thus, we see that $M\to S$ is onto and thus $M=S\oplus S$, which implies $E=\mathcal{O}(a)\oplus\mathcal{O}(b)$ and then it is immediate that $a=0, b=m$ and the sequence splits.