Projective bundle in the quotient convention

Assume the Axiom of Choice as inherited from the relative Proj construction (The Axiom of Choice, Relative Proj of a graded quasi-coherent algebra). Let $S$ be a scheme and let $E$ be a finite locally free ${\mathcal{O}}_S$-module of locally constant rank $r\ge 0$ (Locally free sheaves of finite rank). Write $\mathrm{Sym}(E)={⨁}_{d\ge 0}{\mathrm{Sym}}^d(E)$ for the symmetric algebra of $E$ (Symmetric algebra of a quasi-coherent module): a quasi-coherent graded ${\mathcal{O}}_S$-algebra (Quasi-coherent module on a scheme) with ${\mathrm{Sym}}^0(E)={\mathcal{O}}_S$, ${\mathrm{Sym}}^1(E)=E$, generated as an ${\mathcal{O}}_S$-algebra by its degree-one part $E$, and compatible with base change $S^{\mathrm{\prime }}\to S$ (Symmetric algebras are quasi-coherent and commute with pullback).

Definition. The projective bundle of $E$ over $S$ is the relative Proj $\pi :{\mathbb{P}}_S(E)={\mathrm{Proj}}_S\mathrm{Sym}(E)⟶S,$ with the relative twists ${\mathcal{O}}_{{\mathbb{P}}_S(E)}(n)$, $n\in \mathbb{Z}$ (Relative Proj of a graded quasi-coherent algebra). Since $\mathrm{Sym}(E)$ is generated in degree one, the degree-one part ${\mathrm{Sym}}^1(E)=E$ generates the whole algebra, and there is a canonical surjection ${\pi }^{\ast }E⟶{\mathcal{O}}_{{\mathbb{P}}_S(E)}(1),$ the tautological quotient of the pullback of $E$; it is locally given by the coordinate sections of a projective space, over an open set on which $E\cong {\mathcal{O}}_S^r$.

Quotient convention. This is the quotient (Grothendieck) convention: over an $S$-scheme $g:T\to S$, an $S$-morphism $T\to {\mathbb{P}}_S(E)$ is the same as an isomorphism class of surjections $g^{\ast }E\to L$ with $L$ invertible on $T$, and the universal such quotient is the tautological quotient above; the representing property is proved as Projective bundle represents line quotients. The associated affine construction is $\mathbb{V}(E)={\mathrm{Spec}}_S\mathrm{Sym}(E)$; the rank-zero case below displays its difference from ${\mathbb{P}}_S(E)$ over nonempty $S$.

Rank zero. If $E=0$ then $\mathrm{Sym}(E)={\mathcal{O}}_S$ is concentrated in degree $0$, its irrelevant ideal is $0$, and ${\mathbb{P}}_S(0)=\mathrm{\varnothing }:$ the total space is empty. This is consistent with the quotient convention, because a surjection $0\to L$ onto an invertible sheaf exists only over the empty scheme; the affine bundle $\mathbb{V}(0)=S$ is nonempty whenever $S$ is.

• Frames. If $E{\mathrm{\mid }}_U\cong {\mathcal{O}}_U^r$ over an open $U\subseteq S$ with $r\ge 1$, then $\mathrm{Sym}(E){\mathrm{\mid }}_U\cong {\mathcal{O}}_U[t_1,\dots ,t_r]$ with $\mathrm{deg}{t}_i=1$, so ${\mathbb{P}}_S(E){\times }_{S}U\cong {\mathbb{P}}_U^{r-1}$ by the absolute case of Relative Proj of a graded quasi-coherent algebra; the tautological quotient restricts to the standard quotient ${\mathcal{O}}_U^r\to {\mathcal{O}}_{{\mathbb{P}}_U^{r-1}}(1)$ whose components are the coordinate sections. The case $r=1$ gives a single degree-one generator and is computed on the paired examples page of this pair.
• Base change. For a morphism $S^{\mathrm{\prime }}\to S$ one has ${\mathbb{P}}_{S^{\mathrm{\prime }}}(E_{S^{\mathrm{\prime }}})\cong {\mathbb{P}}_S(E){\times }_S{S}^{\mathrm{\prime }}$, because the symmetric algebra and relative Proj commute with base change (Symmetric algebras are quasi-coherent and commute with pullback, Relative Proj commutes with arbitrary base change).
• Base-change supplier. The construction uses the symmetric algebra and relative Proj only. The base-change bullet above rests on Symmetric algebras are quasi-coherent and commute with pullback and Relative Proj commutes with arbitrary base change. The former supplies the compatibility $f^{\ast }\mathrm{Sym}(\mathcal{F})\cong \mathrm{Sym}(f^{\ast }\mathcal{F})$ and the latter supplies base change for relative Proj; this definition is reconciled with Symmetric algebra of a quasi-coherent module and Relative Proj of a graded quasi-coherent algebra as written.