Let $X=\operatorname{Proj}A$ for some graded ring $A$ such that the irrelevant ideal $A_+$ is generated by the degree one part $A_1$ of $A$.
Then Bosch's $\textit{Algebraic Geometry and Commutative Algebra}$ states on page $427$: taking generators $f_i\in A_1$ of $A_+$ and letting $\eta_{ij}=f_i^n/f_j^n$, this defines a 1-cocycle (with respect to the cover $\{D_+(f_i)\}$), whose associated invertible sheaf $\mathcal{F}$is isomorphic to $\mathcal{O}_X(n)$.
My question is: What does the isomorphism look like? I first thought of gluing the isomorphisms $$\phi_i\colon \mathcal{O}_X(n)\vert_{D_+(f_i)}\cong \mathcal{O}_X\vert_{D_+(f_i)}\cong \mathcal{F}\vert_{D_+(f_i)}$$ but these do not seem to agree on overlaps. So, what's the isomorphism Bosch means? I can't think of any other one.
Thank you.
Think of them as transition functions. Therefore your line bundle can be described (upto isomorphism) as the locally free sheaf of module generated by $f_i^n$ over $D_{+} (f_i)$. But that is equal to $\mathcal {O}_X(n)$.
[ Because, $\mathcal O _X(n) | _{D_{+}(f_i)}= f_i^n p$, where $p \in \mathcal O _X ({D_{+}(f_i)})$. So
$\mathcal O _X(n) | _{D_{+}(f_i f_j)}= f_i^np = f_j^n \left(\frac{f_i^n}{f_j^n}\right)p$. ]