This is exercise 18.3.D of Vakil's Foundations of Algebraic Geometry.
It's to prove that $H^i(\mathbb{P}^n_A, O(m))$ are free, and to compute the dimensions, where $A$ is any commutative ring.
The approach used is to use the Cech cohomology using the open cover $D(x_0), ... D(x_n)$ of $\mathbb{P}^n_A$. First, we consider all of the invertible sheaves at once. We consider $F = \oplus_{m \in \mathbb{Z}} O(m)$ and take the cohomology of this, using the fact that cohomology commutes with direct sum.
For $I \subseteq \{0, ... n\}$, define $U_I$ to be $\cap_{i \in I} D(x_i)$. The next step is to prove that $\Gamma(U_I, F)$ is the Laurent polynomials $A[x_0, ... x_n, \{x_i^{-1} : i \in I \}]$. I understand this part.
Then, we look at how many exponents are allowed to be negative. There are "3 negative exponents" , "2 negative exponents", "1 negative exponent", and "0 negative exponents" cases, where in each case, the first few exponents are required to be negative. Vakil shows that these sequences are exact except for the "3 negative exponents" case, where there is non-exactness at one part. I also understand this part, but I'm not sure as to why it's relevant.
I think that the approach that Vakil is taking is that he's using the "n negative exponents" to give a filtration on the Cech complex, so we can compute the cohomology using a spectral sequence. The part of the proof that shows various sequences are exact shows that the first page is mostly zeroes.
I'm having trouble with writing down what this filtration exactly is. The subquotients must correspond to Laurent polynomials where the first few exponents are required to be negative. Since we're trying to compute cohomology of projective space, I also want this spectral sequence to converge to the $H^i(\mathbb{P}^n_A, O(m))$. How do I do this?