I am not a specialist in spectral sequences and I am trying to apply Beilinson's approach to vector bundles on $\mathbb{P}^1_A$, where $A$ is some good enough ring (e.g. $\mathbb{Z}$ or PID). Where does the proof of Beilinson's theorem about spectral sequences fail when we take as a base of $\mathbb{P}^n$ not a field, but a ring? We first construct the Koszul resolution for the diagonal, apparently this is true for any base (e.g. by direct coordinate calculations using Euler sequence as in the second proof on pg. 65 in https://johncalab.github.io/stuff/beilinson.pdf). After that, we build and calculate hyperdirect images of second projection of our complex, as in the book by Okonek-Schneider-Spindler, which also seems to be independent of the base, and as a result we obtain Beilinson's Theorems.
If this is not true, then is it possible to point out a place where our proof fails for the simplest case – $F$ is an (indecomposable) locally free sheaf of rank 2 on $\mathbb{P}^1_{\mathbb{Z}}$ with $H^1(\mathbb{P}^1_\mathbb{Z},F)=0$.
Or is everything fine and this theorem is also true for rings?