How to complete Vakil's proof that the composition of projective morphisms are projective when the target is quasicompact?

Question

For this question, a morphism $\pi : X \rightarrow Y$ is projective iff there exists a finite type quasicoherent sheaf $\mathcal{E}$ on $Y$ such that $X$ is isomorphic (as a $Y-$scheme) to a closed subscheme of $\mathbb{P}(\mathcal{E})$. I am interested in finding a way to solve this problem as it is presented in Vakil, and not in solving it using completly different methods (such as can be found in e.g. EGA II $5.5.5$, or Stacks Tag $01W7$, although under the additional assumptions that $Y$ is either quasi-separated or Noetherian).

In exercise 17.3.B of Vakil's "Foundations of Algebraic Geometry" notes, he asks to show that if $\pi : X\rightarrow Y$ and $\rho : Y \rightarrow Z$ are projective morphisms and $Z$ is quasi-compact, then $\rho \circ \pi$ is also projective. The hint he gives is to show that in the case where $Z$ is affine, if $\mathcal{L}, \mathcal{M}$ are the very ample line bundles on $X,Y$ coming from pulling back the respective $\mathcal{O}(1)$ bundles from the projective bundles $X$ and $Y$ are closed subschemes of, then there is some $m$ such that $\mathcal{L}\otimes \pi^*(\mathcal{M})^{\otimes m}$ is $\rho\circ \pi-$very ample. He then suggests using that $Z$ is quasicompact to cover it by finitely many open affine pieces, but I can't work out how to use this to prove the result. My first instinct would be to glue together the morphisms constructed over each affine piece, or to extend the construction globally, but in this case neither approach works.

I can see that covering $Z$ by finitely many affine pieces $U_i$ allows us to find a fixed $m$ such that $\mathcal{L}\otimes \pi^*(\mathcal{M})^{\otimes m}$ is $\rho \circ \pi-$relatively very ample upon restriction to each $(\rho \circ \pi)^{-1}(U_i)$, but I don't understand how to use this to show that it is globally $\rho \circ \pi-$relatively very ample. Vakil does mention several times (and later proves) that with locally Noetherian hypotheses, the property of a line bundle on the source being relatively very ample can be checked affine-locally on the target, which would finish the proof if $Z$ was Noetherian, but this isn't part of the hypothesis of $17.3.B$.

My question then, is this:

Is it possible to finish this approach to the exercise, perhaps by showing that $\mathcal{L}\otimes \pi^*(\mathcal{M})^{\otimes m}$ is $\rho \circ \pi-$relatively very ample globally given that it is locally? Or is it really the case that you need to either assume that $Z$ is Noetherian, or take a completely different approach to the proof (such as characterising projective morphisms to quasicompact schemes as those that or both quasiprojective and proper)?

Answer 1

Edit. I decided to just rewrite a proof. I still need quasi-separatedness, however.

Theorem [Stacks, Tag 0C4P]. Suppose $\pi\colon X \to Y$ and $\rho\colon Y \to Z$ are projective morphisms, and $Z$ is quasi-compact and quasi-separated. Then, $\pi \circ \rho$ is projective.

Proof. Let $\mathscr{M}$ be the $\rho$-very ample line bundle on $Y$. Let $X \hookrightarrow \mathbf{P}_Y(\mathscr{E})$ be the closed embedding factoring $\pi$, where $\mathscr{E}$ is a finite type quasicoherent sheaf on $Y$. Now we claim the following:

Key Claim. There exists a finite type quasi-coherent sheaf $\mathscr{G}$ on $Z$ and a surjection $$\rho^*\mathscr{G} \twoheadrightarrow \mathscr{E} \otimes \mathscr{M}^{\otimes m}$$ for $m \gg 0$.

We postpone the proof of the Key Claim for now. Using this surjection, we have a sequence of morphisms $$X \hookrightarrow \mathbf{P}_Y(\mathscr{E}) \cong \mathbf{P}_Y(\mathscr{E} \otimes \mathscr{M}^{\otimes m}) \hookrightarrow \mathbf{P}_Y(\rho^*\mathscr{G})$$ whose composition is still a closed embedding. Moreover, we have an isomorphism $$\mathbf{P}_Y(\rho^*\mathscr{G}) \cong \mathbf{P}_Z(\mathscr{G}) \times_Z Y$$ by [EGAII, 4.1.3.1].

Next, let $Y \hookrightarrow \mathbf{P}_Z(\mathscr{F})$ be the closed embedding factoring $\rho$, where $\mathscr{F}$ is a finite type quasicoherent sheaf on $Z$. Then, we have a closed embedding $$X \hookrightarrow \mathbf{P}_Z(\mathscr{G}) \times_Z Y \hookrightarrow \mathbf{P}_Z(\mathscr{G}) \times_Z \mathbf{P}_Z(\mathscr{F})$$ and composing by the (relative) Segre embedding [EGAII, §4.3], we get a closed embedding $$X \hookrightarrow \mathbf{P}_Z(\mathscr{G} \otimes \mathscr{F})$$ Since $\mathscr{G}$ and $\mathscr{F}$ were finite type quasi-coherent sheaves on $Z$, we have that $\pi \circ \rho$ is indeed projective. $\blacksquare$

We now return to the proof of the Key Claim. This is where we use that $Z$ is quasi-separated.

Proof of Key Claim. Since projective morphisms are proper, we can apply [Vakil, 17.3.9] to say that $\mathscr{M}$ is in fact $\rho$-ample, and so for $m \gg 0$, we have that $\mathscr{E} \otimes \mathscr{M}^{\otimes m}$ is $\rho$-globally generated, that is, we have that the canonical map $$\rho^*\rho_*\!\left(\mathscr{E} \otimes \mathscr{M}^{\otimes m}\right) \twoheadrightarrow \mathscr{E} \otimes \mathscr{M}^{\otimes m}$$ is a surjection. By [Görtz–Wedhorn, 10.50] we can write $$\rho_*\!\left(\mathscr{E} \otimes \mathscr{M}^{\otimes m}\right) = \varinjlim \mathscr{G}_\lambda$$ for the filtered system of finite type quasi-coherent subsheaves $\mathscr{G}_\lambda \subset \rho_*\!\left(\mathscr{E} \otimes \mathscr{M}^{\otimes m}\right)$. Since $\rho^*$ is the left adjoint of $\rho_*$, it preserves colimits, and the surjection above becomes a surjection $$\varinjlim \rho^*\mathscr{G}_\lambda \twoheadrightarrow \mathscr{E} \otimes \mathscr{M}^{\otimes m}$$ and by [Görtz–Wedhorn, 10.47], for $\lambda$ large enough, we have a surjection $$\rho^*\mathscr{G}_\lambda \twoheadrightarrow \mathscr{E} \otimes \mathscr{M}^{\otimes m}$$ as desired. $\blacksquare$

Remark. Here are possible ideas for getting rid of the quasi-separatedness assumption:

1. Try using the affine case. Since $\mathscr{M}$ is ample, we have locally have surjections $$\rho^*\mathcal{O}_{U_i}^{\oplus n_i} \twoheadrightarrow \left.\left(\mathscr{E} \otimes \mathscr{M}^{\otimes m_i} \right)\right\rvert_{\rho^{-1}(U_i)}$$ on each element $U_i$ of a finite open affine cover of $Z$. Then, we could hope to glue these surjections together somehow. One method is to extend the sheaves on the left-hand side to finite type quasi-coherent sheaves on all of $Z$, and use [EGAInew, 6.9.10.1], but that still uses quasi-separatedness. The issue is that extension theorems for finite type quasi-coherent sheaves need quasi-separatedness to make their glueing arguments work; see [EGAInew, 6.9; Görtz–Wedhorn, §10.11; Stacks, Tag 01PD].
2. In the proof above, to apply [Görtz–Wedhorn, 10.47], all we needed was to write $\rho_*\!\left(\mathscr{E} \otimes \mathscr{M}^{\otimes m}\right)$ as a filtered colimit of finite type quasi-coherent sheaves. Perhaps this can be done in this case even without quasi-separatedness.

Remark. The Stacks Project [Stacks, Tag 0C4P] also restricts to $Z$ quasi-separated.