The Stacks project

Proof. By Lemma 31.28.4 the Picard groups of the opens $D_+(T_ i) \cong \mathbf{A}^ n_ R$ are trivial. Thus if $\mathcal{L}$ is an invertible module on $\mathbf{P}^ n_ R$, then it is given by a $1$-cocycle with values in the sheaf of invertible functors $\mathcal{O}^*$ for the open covering

Observe that for $i \not= j$ we have

Thus such a cocycle $(g_{ij})$ is given by units

with $u_{ij} \in R^*$ and $e_{ij} \in \mathbf{Z}$ satisfying the cocycle condition. The cocycle condition over $D_+(T_ iT_ jT_ k)$ for $\# \{ i, j, k\} = 3$ tell us that

Whence $u_{ij} = u_{i1} u_{j1}^{-1}$ is a boundary. Thus all isomorphism classes of invertible modules are given by taking the cocycle with

for some $e \in \mathbf{Z}$. Since $\mathcal{O}(n)$ has trivializing section $T_ i^ n$ over $D_+(T_ i)$ we see that the corresponding cocycle of $\mathcal{O}(n)$ is $(T_ i/T_ j)^ n$ and the proof is complete. $\square$