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.
Comments (0)