Lemma 70.12.2. 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

In addition we have

with obvious notation.

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