The Stacks project

Lemma 37.51.1. Let $R$ be a ring. Let $P$ be a proper scheme over $R$ and let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ P$-module. Set $A = \bigoplus _{m \geq 0} \Gamma (P, \mathcal{L}^{\otimes m})$. Then $P = \text{Proj}(A)$ and diagram ( becomes the diagram

\[ \xymatrix{ \underline{\mathop{\mathrm{Spec}}}_ P \left( \bigoplus \nolimits _{m \in \mathbf{Z}} \mathcal{L}^{\otimes m} \right) \ar[r] \ar@{=}[d] & L = \underline{\mathop{\mathrm{Spec}}}_ P \left( \bigoplus \nolimits _{m \geq 0} \mathcal{L}^{\otimes m} \right) \ar[d]^\sigma \ar[r]_-\pi & P \ar[d] \\ U \ar[r] & X \ar[r] & Z } \]

having the properties explained above.

Proof. We have $P = \text{Proj}(A)$ by Morphisms, Lemma 29.43.17. Moreover, by Properties, Lemma 28.28.2 via this identification we have $\mathcal{O}_ P(m) = \mathcal{L}^{\otimes m}$ for all $m \in \mathbf{Z}$. $\square$

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