Lemma 27.18.2. Let $S$ be a scheme. Let $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ be quasi-coherent graded $\mathcal{O}_ S$-algebras. Set $X = \underline{\text{Proj}}_ S(\mathcal{A})$, $Y = \underline{\text{Proj}}_ S(\mathcal{B})$ and $Z = \underline{\text{Proj}}_ S(\mathcal{C})$. Let $\varphi : \mathcal{A} \to \mathcal{B}$, $\psi : \mathcal{B} \to \mathcal{C}$ be graded $\mathcal{O}_ S$-algebra maps. Then we have

In addition we have

with obvious notation.

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