The Stacks project

Proof. Affine locally this reduces to the following statement: Let $R$ be a ring and let $A$ be a finitely presented graded $R$-algebra. Then $\text{Proj}(A) \to \mathop{\mathrm{Spec}}(R)$ is of finite presentation and $\mathcal{O}_{\text{Proj}(A)}(d)$ is a $\mathcal{O}_{\text{Proj}(A)}$-module of finite presentation. The finite presentation condition implies we can choose a presentation

where $R[X_1, \ldots , X_ n]$ is a polynomial ring graded by giving weights $d_ i$ to $X_ i$ and $F_1, \ldots , F_ m$ are homogeneous polynomials of degree $e_ j$. Let $R_0 \subset R$ be the subring generated by the coefficients of the polynomials $F_1, \ldots , F_ m$. Then we set $A_0 = R_0[X_1, \ldots , X_ n]/(F_1, \ldots , F_ m)$. By construction $A = A_0 \otimes _{R_0} R$. Thus by Constructions, Lemma 27.11.6 it suffices to prove the result for $X_0 = \text{Proj}(A_0)$ over $R_0$. By Lemma 31.30.2 we know $X_0$ is of finite type over $R_0$ and $\mathcal{O}_{X_0}(d)$ is a quasi-coherent $\mathcal{O}_{X_0}$-module of finite type. Since $R_0$ is Noetherian (as a finitely generated $\mathbf{Z}$-algebra) we see that $X_0$ is of finite presentation over $R_0$ (Morphisms, Lemma 29.21.9) and $\mathcal{O}_{X_0}(d)$ is of finite presentation by Cohomology of Schemes, Lemma 30.9.1. This finishes the proof. $\square$