Lemma 27.11.2 (01MZ)—The Stacks project

Table of contents
Part 2: Schemes
Chapter 27: Constructions of Schemes
Section 27.11: Functoriality of Proj
Lemma 27.11.2 (cite)

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

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

In addition we have

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

with obvious notation.

Proof. Omitted. $\square$

Comments (1)

Comment #8511 by Jordan on June 19, 2023 at 11:47

Shouldn't it be $U(\psi\circ\varphi)=r_{\psi}^{-1}(U(\varphi))$ ? I'd expect to obtain $U(\psi\circ\varphi)$ by first forming the open $U(\varphi)$ in $Y$ , and then pulling it back to $Z$ to form $r_{\psi}^{-1}(U(\varphi))= U(\psi)\times_{r_{\psi},Y}U(\varphi)$ .

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