Lemma 31.30.4. 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}. The following conditions are equivalent
\mathcal{A}_0 is a finite type \mathcal{O}_ S-module and \mathcal{A} is of finite type as an \mathcal{A}_0-algebra,
\mathcal{A}_0 is a finite type \mathcal{O}_ S-module and \mathcal{A} is of finite type as an \mathcal{O}_ S-algebra
If these conditions hold, then p is locally projective and in particular proper.
Proof.
Assume that \mathcal{A}_0 is a finite type \mathcal{O}_ S-module. Choose an affine open U = \mathop{\mathrm{Spec}}(R) \subset X such that \mathcal{A} corresponds to a graded R-algebra A with A_0 a finite R-module. Condition (1) means that (after possibly localizing further on S) that A is a finite type A_0-algebra and condition (2) means that (after possibly localizing further on S) that A is a finite type R-algebra. Thus these conditions imply each other by Algebra, Lemma 10.6.2.
A locally projective morphism is proper, see Morphisms, Lemma 29.43.5. Thus we may now assume that S = \mathop{\mathrm{Spec}}(R) and X = \text{Proj}(A) and that A_0 is finite over R and A of finite type over R. We will show that X = \text{Proj}(A) \to \mathop{\mathrm{Spec}}(R) is projective. We urge the reader to prove this for themselves, by directly constructing a closed immersion of X into a projective space over R, instead of reading the argument we give below.
By Lemma 31.30.2 we see that X is of finite type over \mathop{\mathrm{Spec}}(R). Constructions, Lemma 27.10.6 tells us that \mathcal{O}_ X(d) is ample on X for some d \geq 1 (see Properties, Section 28.26). Hence X \to \mathop{\mathrm{Spec}}(R) is quasi-projective (by Morphisms, Definition 29.40.1). By Morphisms, Lemma 29.43.12 we conclude that X is isomorphic to an open subscheme of a scheme projective over \mathop{\mathrm{Spec}}(R). Therefore, to finish the proof, it suffices to show that X \to \mathop{\mathrm{Spec}}(R) is universally closed (use Morphisms, Lemma 29.41.7). This follows from Lemma 31.30.3.
\square
Comments (0)