Lemma 29.44.16 (0B3I)—The Stacks project

Lemma 29.44.16. A finite morphism is projective.

Proof. Let $f : X \to S$ be a finite morphism. Then $f_*\mathcal{O}_ X$ is a quasi-coherent $\mathcal{O}_ S$-module (Lemma 29.11.5) of finite type (by our definition of finite morphisms and Properties, Lemma 28.16.1). We claim there is a closed immersion

\[ \sigma : X \longrightarrow \mathbf{P}(f_*\mathcal{O}_ X) = \underline{\text{Proj}}_ S(\text{Sym}^*_{\mathcal{O}_ S}(f_*\mathcal{O}_ X)) \]

over $S$, which finishes the proof. Namely, we let $\sigma $ be the morphism which corresponds (via Constructions, Lemma 27.16.11) to the surjection

\[ f^*f_*\mathcal{O}_ X \longrightarrow \mathcal{O}_ X \]

coming from the adjunction map $f^*f_* \to \text{id}$. Then $\sigma $ is a closed immersion. Namely, affine locally on $S$ we can write $X = \mathop{\mathrm{Spec}}(A)$ and $S = \mathop{\mathrm{Spec}}(R)$. Since $X$ is finite over $S$ we may choose $a_1, \ldots , a_ n \in A$ generating $A$ as an $R$-module. Then the $R$-algebra map $R[T_0, T_1, \ldots , T_ n] \to \text{Sym}_ R^*(A)$ sending $T_0$ to $1$ and $T_ i$ to $a_ i$ for $1 \leq i \leq n$ is surjective. Whence $\mathbf{P}(f_*\mathcal{O}_ X)$ is a closed subscheme of $\mathbf{P}^ n_ S$ by Constructions, Lemma 27.11.3. Thus it suffices to prove that the induced morphism $X \to \mathbf{P}^ n_ S$ is a closed immersion (see for example, Lemma 29.2.2). The reader checks that $X \to \mathbf{P}^ n_ S$ has image contained in the open $D_+(T_0) \cong \mathbf{A}^ n_ S$ and that $X \to \mathbf{A}^ n_ S$ corresponds to the surjective $R$-algebra map $R[x_1, \ldots , x_ n] \to \text{Sym}_ R^*(A)$ sending $x_ i$ to $a_ i$. Whence $X \to \mathbf{P}^ n_ S$ is an immersion (as a composition of a closed immersion and an open immersion). Since $X$ is finite over $S$ the image of this immersion is closed (this uses Lemma 29.44.11, Lemma 29.41.7, and Constructions, Lemma 27.13.4). We conclude by Schemes, Lemma 26.10.4. $\square$

Comments (10)

Comment #1835 by Pieter Belmans on February 08, 2016 at 13:02

In order to apply to tag 27.16.11 shouldn't the graded algebra structure be made more explicit? In particular: one puts $\mathcal{O}_S$ in degree 0 and $f_*(\mathcal{O}_X)$ in degrees $\geq 1$ , and then one can appeal to 27.16.11. I might be overlooking something, but right now I don't see how we can use the referenced lemma.

Comment #1872 by Johan on April 01, 2016 at 13:41

Yes, I have now added the explicit form of the projective bundle as the relative Proj of the symmetric algebra over $\mathcal{O}_S$ of $f_*\mathcal{O}_X$ . I think this will alleviate your concerns, see here.

Comment #7118 by Arnab Kundu on March 16, 2022 at 12:51

Maybe a silly point: but why is the map $\sigma$ a closed immersion? I understand that the graph of $\sigma$ is a closed immersion by Lemma O1KS and Lemma 01OD. Am I missing something?

Comment #7119 by Johan on March 16, 2022 at 13:54

Not silly! The reference should have been to Lemma 26.21.11. Will fix this later.

Comment #7283 by Johan on May 03, 2022 at 13:18

Thanks and fixed here.

Comment #8444 by Johan on May 31, 2023 at 15:21

Argh, the reference to Lemma 26.21.11 is still wrong because the Proj is over $S$ and not over $X$ . The best thing would be for people to work out for themselves why $\sigma$ is a closed immersion (by doing a local computation). But we can also see it using theory. Write $X$ as $\text{Proj}_S(\mathcal{A})$ where $\mathcal{A} = (f_*\mathcal{O}_X)[T]$ . (Namely, we have $\text{Proj}(S[T]) = \text{Spec}(S)$ and there is a relative version of this.) Then with $\mathcal{B} = \text{Sym}^*_{\mathcal{O}_S}(f_*\mathcal{O}_X)$ there is a map $\psi : \mathcal{B} \to \mathcal{A}$ of graded $\mathcal{O}_S$ -algebras. (Namely, given a ring map $R \to S$ there is a graded $R$ -algebra map $\text{Sym}_R^*(S) \to S[T]$ , $s \mapsto sT$ and there is a relative version of this.) The map $\psi$ is surjective in all sufficiently high degrees. (Namely, in degrees $\geq 1$ .) Whence the corresponding morphism $r_\psi : X \to \mathbf{P}(f_*\mathcal{O}_X)$ is everywhere defined and a closed immersion, see Lemma 27.18.3.

Comment #8447 by Jason Starr on June 01, 2023 at 06:19

By the discussion in "N'eron models" around finite flat morphisms, after a faithfully flat base change there is a filtration of f_*O_X whose associated graded pieces are f-pushforwards of pushforwards of invertible O_S-modules along sections of f (i.e., multisections are "scheme-theoretic unions" of sections after faithfully flat base change). The composition of those sections with sigma give a corresponding filtration of the structure sheaf of the relative Proj (by ideal sheaves) compatible with the filtration on the pushforward of O_X. You can use this to prove that the morphism from the structure sheaf of the relative Proj to the pushforward of O_X is surjective.

Comment #8450 by Aise Johan de Jong on June 01, 2023 at 10:57

Thanks Jason!

Comment #8451 by Jason Starr on June 01, 2023 at 13:53

I think my comment was wrong. There is a filtration after base change, and you can use it to prove the result. However, that is not the shortest proof, and I do not know if the filtration comes from a sequence of sections. I need to double-check what is in "N'eron models" -- they have effectively evicted us from our offices while they repair the HVAC system.

Comment #9066 by Stacks project on May 06, 2024 at 13:31

OK, I finally fixed the closed immersion thing by just proving it affine locally. See here.

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