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The Stacks project

Lemma 31.30.7. 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} is a finitely presented \mathcal{O}_ S-algebra, then p is of finite presentation and \mathcal{O}_ X(d) is an \mathcal{O}_ X-module of finite presentation.

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

A = R[X_1, \ldots , X_ n]/(F_1, \ldots , F_ m)

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


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