Let $A$ be a graded ring satisfying the usual finiteness conditions of EGA II (for example $A_0$ is noetherian, $A_1$ is finite over $A_0$ and $A$ is generated by finitely many elements of $A_1$ as an $A_0$-algebra), and consider the projective scheme $P:=\mathrm{Proj}(A)$. Then every coherent sheaf $F$ on $P$ may be written as the cokernel of some map of the form $\delta : \oplus_j \mathcal{O}(m_j) \to \oplus_i \mathcal{O}(n_i)$, which can also be seen as a matrix of homogeneous elements of certain degrees. I would like to call this a (twisted) presentation of $F$.
My main question is: How can we explicitly compute $H^0(P,F)$ in terms of $\delta$? If $\delta$ is injective, we can use the long exact sequence of cohomology groups since we know the cohomology groups of the Serre twists. But what about the general case? For my purposes it would be enough to write $H^0(P,F)$ as an extension of known modules.
Here is what I've done so far: Since we can present $\mathcal{O}$ with sufficiently "small" twists, it suffices to compute $H^0(P,F(k))$ for large enough $k$. If $K$ denotes the kernel of $\oplus_i \mathcal{O}(n_i) \to M$, we have $H^1(P,K(k))=0$ for large enough $k$, so by changing notation we may assume $H^1(P,K)=0$, which means that $0 \to H^0(P,K) \to H^0(P,\oplus_i \mathcal{O}(n_i)) \to H^0(P,F) \to 0$ is exact. Therefore it suffices to describe $H^0(P,K)$. By induction and some other short exact sequences it is enough to consider the case $K \subseteq \mathcal{O}(n)$ for some $n \geq 0$. What can we say about $H^0(P,K)$ then?
My background is that I would like to prove the Main Theorem of Artin-Zhang's paper on noncommutative projective schemes for abelian $\otimes$-categories, instead of abelian categories, which seems quite natural.
