Lemma 27.20.1. With notation $S$, $\mathcal{A}$, $\mathcal{L}$ and $\mathcal{B}$ as above. There is a canonical isomorphism

with the following properties

There are isomorphisms $\theta _ n : g^*\mathcal{O}_{P'}(n) \to \mathcal{O}_ P(n) \otimes \pi ^*\mathcal{L}^{\otimes n}$ which fit together to give an isomorphism of $\mathbf{Z}$-graded algebras

\[ \theta : g^*\left( \bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{O}_{P'}(n) \right) \longrightarrow \bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{O}_ P(n) \otimes \pi ^*\mathcal{L}^{\otimes n} \]For every open $V \subset S$ the diagrams

\[ \xymatrix{ \mathcal{A}_ n(V) \otimes \mathcal{L}^{\otimes n}(V) \ar[r]_{multiply} \ar[d]^{\psi \otimes \pi ^*} & \mathcal{B}_ n(V) \ar[dd]^\psi \\ \Gamma (\pi ^{-1}V, \mathcal{O}_ P(n)) \otimes \Gamma (\pi ^{-1}V, \pi ^*\mathcal{L}^{\otimes n}) \ar[d]^{multiply} \\ \Gamma (\pi ^{-1}V, \mathcal{O}_ P(n) \otimes \pi ^*\mathcal{L}^{\otimes n}) & \Gamma (\pi '^{-1}V, \mathcal{O}_{P'}(n)) \ar[l]_-{\theta _ n} } \]are commutative.

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