The Stacks project

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.

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.

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?

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

Thanks and fixed here.

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.

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.

Thanks Jason!

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.