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The Stacks project

Lemma 71.12.2. Let S be a scheme. Let X be an algebraic space over S. Let \mathcal{A}, \mathcal{B}, and \mathcal{C} be quasi-coherent graded \mathcal{O}_ X-algebras. Set P = \underline{\text{Proj}}_ X(\mathcal{A}), Q = \underline{\text{Proj}}_ X(\mathcal{B}) and R = \underline{\text{Proj}}_ X(\mathcal{C}). Let \varphi : \mathcal{A} \to \mathcal{B}, \psi : \mathcal{B} \to \mathcal{C} be graded \mathcal{O}_ X-algebra maps. Then we have

U(\psi \circ \varphi ) = r_\varphi ^{-1}(U(\psi )) \quad \text{and} \quad r_{\psi \circ \varphi } = r_\varphi \circ r_\psi |_{U(\psi \circ \varphi )}.

In addition we have

\theta _\psi \circ r_\psi ^*\theta _\varphi = \theta _{\psi \circ \varphi }

with obvious notation.

Proof. Omitted. \square


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