Geometric vector bundle with the sections convention

Assume the Axiom of Choice, inherited from the relative-spectrum and symmetric-algebra machinery used below (The Axiom of Choice, Glue relative spectra of affine-local algebras, Symmetric algebras are quasi-coherent and commute with pullback). Let $X$ be a scheme (Schemes and morphisms over a base).

Graded coordinate algebras. Let $\pi :V\to X$ be an affine $X$-scheme (Affine morphisms). Its relative coordinate algebra is $\mathcal{A}={\pi }_{\ast }{\mathcal{O}}_V,$ a sheaf of commutative unital ${\mathcal{O}}_X$-algebras with structure map the comorphism ${\pi }^{\mathrm{♯}}:{\mathcal{O}}_X\to {\pi }_{\ast }{\mathcal{O}}_V$ (Direct image of a sheaf along a continuous map, Morphisms of schemes). By the relative-spectrum characterization of affine morphisms, $\pi$ is recovered from the affine-locally module-associated algebra $\mathcal{A}$ (Affine morphisms are relative spectra, Affine-local quasi-coherent algebras before general sheaf theory).

A grading on a sheaf $\mathcal{A}$ of commutative unital ${\mathcal{O}}_X$-algebras is a family $({\mathcal{A}}_d{)}_{d\ge 0}$ of ${\mathcal{O}}_X$-submodules with ${\mathcal{A}}_d{\mathcal{A}}_e\subseteq {\mathcal{A}}_{d+e}$ for all $d,e\ge 0$ whose multiplication maps assemble into an isomorphism of ${\mathcal{O}}_X$-modules ${⨁}_{d\ge 0}{\mathcal{A}}_d\to \mathcal{A}$. The grading is normalised when ${\mathcal{A}}_0={\mathcal{O}}_X$, the structure copy of the algebra; this is the only case used below. An element of ${\mathcal{A}}_d$ is homogeneous of degree $d$. A morphism of graded ${\mathcal{O}}_X$-algebras is an ${\mathcal{O}}_X$-algebra morphism $\phi :\mathcal{A}\to \mathcal{B}$ with $\phi ({\mathcal{A}}_d)\subseteq {\mathcal{B}}_d$ for every $d\ge 0$; if $\phi$ is bijective, then $\phi ({\mathcal{A}}_d)={\mathcal{B}}_d$ for every $d$: an element of ${\mathcal{B}}_d$ is a finite sum of elements of the graded pieces $\phi ({\mathcal{A}}_e)\subseteq {\mathcal{B}}_e$, and comparing components inside the direct sum ${⨁}_e{\mathcal{B}}_e$ shows that it lies in $\phi ({\mathcal{A}}_d)$; hence the inverse is graded as well.

Geometric vector bundles. A geometric vector bundle over $X$, in the sections convention, is an affine $X$-scheme $\pi :V\to X$ together with a normalised grading $\mathcal{A}={⨁}_{d\ge 0}{\mathcal{A}}_d$ on its relative coordinate algebra $\mathcal{A}={\pi }_{\ast }{\mathcal{O}}_V$ such that there are an open cover $X={\bigcup }_i{U}_i$, integers $r_i\ge 0$, and isomorphisms of graded ${\mathcal{O}}_{U_i}$-algebras $\mathcal{A}{\mathrm{\mid }}_{U_i}\cong \mathrm{Sym}\left({\mathcal{O}}_{U_i}^{r_i}\right),$ where the symmetric algebra carries its grading generated in degree one (Symmetric algebra of a quasi-coherent module, Symmetric algebras are quasi-coherent and commute with pullback).

The rank. On a trivializing open $U_i$ the degree-one part of such an isomorphism is an isomorphism ${\mathcal{A}}_1{\mathrm{\mid }}_{U_i}\cong {\mathrm{Sym}}^1({\mathcal{O}}_{U_i}^{r_i})={\mathcal{O}}_{U_i}^{r_i}$, so ${\mathcal{A}}_1$ is locally free of finite rank and has rank $r_i$ on $U_i$ (Symmetric algebra of a quasi-coherent module, Locally free sheaves of finite rank). On an overlap $U_i\cap U_j$ two trivializations force $r_i=r_j$ pointwise, by the well-definedness of the rank of a finite locally free sheaf (Locally free sheaves of finite rank); the $r_i$ therefore glue to a locally constant function $r:X\to \mathbb{N}$, called the rank of the geometric vector bundle. Rank $0$ is allowed: it means ${\mathcal{A}}_1=0$, hence $\mathcal{A}={\mathcal{O}}_X$ and $V=X$.

Linearity of the transitions. Let $W\subseteq X$ be open and let $\mathcal{A}{\mathrm{\mid }}_W\cong \mathrm{Sym}({\mathcal{O}}_W^r)$ and $\mathcal{A}{\mathrm{\mid }}_W\cong \mathrm{Sym}({\mathcal{O}}_W^s)$ be two trivializations. Their comparison is an isomorphism $\psi :\mathrm{Sym}({\mathcal{O}}_W^r)\to \mathrm{Sym}({\mathcal{O}}_W^s)$ of graded ${\mathcal{O}}_W$-algebras, so the degree-one part $\theta ={\psi }_1$ is an isomorphism ${\mathcal{O}}_W^r\to {\mathcal{O}}_W^s$ and $\psi =\mathrm{Sym}(\theta )$: both maps are graded algebra morphisms out of $\mathrm{Sym}({\mathcal{O}}_W^r)$ with the same restriction to the degree-one generators, and that algebra is generated in degree one and determined by the universal property of Symmetric algebra of a quasi-coherent module. Thus any two trivializations of a geometric vector bundle differ by an invertible linear map of degree one, which is the transition condition of the classical definition; conversely an atlas of trivializations whose transition isomorphisms are linear transports the standard gradings to a well-defined grading on $\mathcal{A}$. The grading belongs to the data of a geometric vector bundle and is required to be preserved by morphisms, so morphisms of geometric vector bundles are not arbitrary $X$-morphisms.

Total space of a finite locally free module. Let $\mathcal{E}$ be a finite locally free ${\mathcal{O}}_X$-module, with rank function $r$ (Locally free sheaves of finite rank). Its dual ${\mathcal{E}}^{\vee }$ is finite locally free of the same rank, and ${\mathcal{E}}^{\vee }{\mathrm{\mid }}_U\cong {\mathcal{O}}_U^r$ on every open $U$ on which $\mathcal{E}{\mathrm{\mid }}_U\cong {\mathcal{O}}_U^r$ (Dual and base change for finite locally free sheaves). Put ${\mathcal{A}}_{\mathcal{E}}:=\mathrm{Sym}({\mathcal{E}}^{\vee }),$ a quasi-coherent graded ${\mathcal{O}}_X$-algebra whose degree-one part is ${\mathcal{E}}^{\vee }$, with restriction ${\mathcal{A}}_{\mathcal{E}}{\mathrm{\mid }}_U\cong \mathrm{Sym}({\mathcal{O}}_U^r)\cong {\mathcal{O}}_U[T_1,\dots ,T_r]$ on every such chart (Symmetric algebra of a quasi-coherent module, Symmetric algebras are quasi-coherent and commute with pullback). More generally, for an arbitrary affine open $U=\mathrm{Spec}R\subseteq X$ the module ${\mathcal{E}}^{\vee }{\mathrm{\mid }}_U$ is quasi-coherent, because ${\mathcal{E}}^{\vee }$ is locally free (Locally free sheaves of finite rank, Quasi-coherent module on a scheme), so ${\mathcal{E}}^{\vee }{\mathrm{\mid }}_U\cong \tilde{M}$ for an $R$-module $M$, and then the affine model of the symmetric algebra gives ${\mathcal{A}}_{\mathcal{E}}{\mathrm{\mid }}_U\cong ({\mathrm{Sym}}_{R}M{)}^{\sim }$, the module-associated sheaf with its usual principal-open localizations (Symmetric algebra of a quasi-coherent module). Hence ${\mathcal{A}}_{\mathcal{E}}$ is affine-locally module-associated (Affine-local quasi-coherent algebras before general sheaf theory). The relative spectrum $\mathbb{V}(\mathcal{E}):={\mathrm{Spec}}_X{\mathcal{A}}_{\mathcal{E}}⟶X$ exists, is affine over $X$, has relative coordinate algebra canonically isomorphic to ${\mathcal{A}}_{\mathcal{E}}$, and carries the grading of ${\mathcal{A}}_{\mathcal{E}}$; it is therefore a geometric vector bundle of rank $r$, the geometric total space of $\mathcal{E}$ (Glue relative spectra of affine-local algebras, Affine morphisms are relative spectra). On a trivializing chart $\mathbb{V}(\mathcal{E})$ restricts to ${\mathrm{Spec}}_U\mathrm{Sym}({\mathcal{O}}_U^r)\cong {\mathbf{A}}_U^r$, the relative affine space of Schemes and morphisms over a base; in particular $\mathbb{V}({\mathcal{O}}_X^r)\cong {\mathbf{A}}_X^r$. The degree-one part of ${\mathcal{A}}_{\mathcal{E}}$ is ${\mathcal{E}}^{\vee }$, so $\mathcal{E}\cong ({\mathcal{A}}_{\mathcal{E}}{)}_1^{\vee }$ by the double-dual isomorphism (Dual and base change for finite locally free sheaves): this is the sense in which the symmetric algebra is formed on the dual and $\mathcal{E}$ is regarded as the sheaf of sections, the sections convention fixed by this item.

Rank zero. For $\mathcal{E}=0$ one has ${\mathcal{E}}^{\vee }=0$ and $\mathrm{Sym}(0)={\mathcal{O}}_X$, since the degree-one part vanishes (Symmetric algebra of a quasi-coherent module). Hence $\mathbb{V}(0)={\mathrm{Spec}}_X{\mathcal{O}}_X=X,$ the identity $X$-scheme, of rank $0$: the zero geometric vector bundle, whose total space is the base with one zero vector over every point.

Morphisms. Let $(V,\mathcal{A})$ and $(W,\mathcal{B})$ be geometric vector bundles over $X$ with structure morphisms ${\pi }_V$ and ${\pi }_W$. A morphism of geometric vector bundles $\phi :V\to W$ is an $X$-morphism (Schemes and morphisms over a base) whose comorphism ${\phi }^{\mathrm{♯}}:{\mathcal{O}}_W\to {\phi }_{\ast }{\mathcal{O}}_V$ (Morphisms of schemes) induces, by pushforward along ${\pi }_W$ and the identity ${\pi }_W\circ \phi ={\pi }_V$, a morphism of graded ${\mathcal{O}}_X$-algebras ${\phi }^{\mathrm{♯}}:\mathcal{B}={\pi }_{W\ast }{\mathcal{O}}_W⟶({\pi }_W\circ \phi {)}_{\ast }{\mathcal{O}}_V={\pi }_{V\ast }{\mathcal{O}}_V=\mathcal{A}\mathrm{.}$ Identities and composites of such morphisms are again such, because the identity is graded and a composite of graded algebra morphisms is graded, so geometric vector bundles over $X$ with these morphisms form a category; an isomorphism is an invertible morphism in this category.

Restriction to an open subscheme. For an open subscheme $W\subseteq X$, the base change $V{\times }_{X}W\to W$ with the restricted grading on $\mathcal{A}{\mathrm{\mid }}_W$ is again a geometric vector bundle over $W$, of the restricted rank: the restriction clause of Glue relative spectra of affine-local algebras identifies the relative spectrum over $W$ with the base change, and $\mathrm{Sym}({\mathcal{E}}^{\vee }){\mathrm{\mid }}_W\cong \mathrm{Sym}(({\mathcal{E}}^{\vee }){\mathrm{\mid }}_W)$ is the restriction isomorphism of Symmetric algebras are quasi-coherent and commute with pullback.

Choice. The Axiom of Choice is used only through the relative-spectrum construction and the symmetric-algebra construction cited above; the data of a geometric vector bundle are a scheme morphism and a grading, and the construction of $\mathbb{V}(\mathcal{E})$ uses the family of all affine opens trivializing $\mathcal{E}$ without selecting a chart or an isomorphism (The Axiom of Choice).