Lemma 31.30.6 (0D4C)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.30: Relative Proj
Lemma 31.30.6 (cite)

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Lemma 31.30.6. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$ -algebra. Let $p : X = \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$ . If $\mathcal{A}_ d$ is a flat $\mathcal{O}_ S$ -module for $d \gg 0$ , then $p$ is flat and $\mathcal{O}_ X(d)$ is flat over $S$ .

Proof. Affine locally flatness of $X$ over $S$ reduces to the following statement: Let $R$ be a ring, let $A$ be a graded $R$ -algebra with $A_ d$ flat over $R$ for $d \gg 0$ , let $f \in A_ d$ for some $d > 0$ , then $A_{(f)}$ is flat over $R$ . Since $A_{(f)} = \mathop{\mathrm{colim}}\nolimits A_{nd}$ where the transition maps are given by multiplication by $f$ , this follows from Algebra, Lemma 10.39.3. Argue similarly to get flatness of $\mathcal{O}_ X(d)$ over $S$ . $\square$

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