Lemma 27.11.2. Let A, B, and C be graded rings. Set X = \text{Proj}(A), Y = \text{Proj}(B) and Z = \text{Proj}(C). Let \varphi : A \to B, \psi : B \to C be graded ring maps. Then we have
U(\psi \circ \varphi ) = r_\psi ^{-1}(U(\varphi )) \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.
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