Lemma 27.11.6 (01N2)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 27: Constructions of Schemes
Section 27.11: Functoriality of Proj
Lemma 27.11.6 (cite)

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Lemma 27.11.6. With hypotheses and notation as in Lemma 27.11.1 above. Assume there exists a ring map $R \to A_0$ and a ring map $R \to R'$ such that $B = R' \otimes _ R A$. Then

$U(\psi ) = Y$,
the diagram

\[ \xymatrix{ Y = \text{Proj}(B) \ar[r]_{r_\psi } \ar[d] & \text{Proj}(A) = X \ar[d] \\ \mathop{\mathrm{Spec}}(R') \ar[r] & \mathop{\mathrm{Spec}}(R) } \]

is a fibre product square, and
the maps $\theta : r_\psi ^*\mathcal{O}_ X(n) \to \mathcal{O}_ Y(n)$ are isomorphisms.

Proof. This follows immediately by looking at what happens over the standard opens $D_{+}(f)$ for $f \in A_{+}$. $\square$

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