Proposition
Let \(\mathcal L\) be an invertible sheaf on \(S\) a scheme, \(\mathcal A\) some sheaf of graded \(\mathcal O_S\)-algebras and set \[\mathcal A_{\mathcal L} := \bigoplus_{d\geq 0}\mathcal A_d\otimes \mathcal L^{\otimes d}.\] Set \(X_{\mathcal L} = \Proj \mathcal A_{\mathcal L}\) and \(X = \Proj \mathcal A\) with structure morphisms \(\pi':X_{\mathcal L} \to S\) and \(\pi:X\to S\) respectively. Then there is an isomorphism of \(S\)-schemes \[g_{\mathcal L}:X_{\mathcal L}\xrightarrow{\sim} X\] such that \(\mathcal O_{X_{\mathcal L}}(n) = g^*_{\mathcal L}(\mathcal O_X(n))\otimes \pi'^*(\mathcal L^{\otimes n})\) for all \(n\in \mathbb Z\).