The rank-zero bundle

Given: A scheme $X$ and the zero ${\mathcal{O}}_X$-module $0$.

Finite local freeness and rank (Locally free sheaves of finite rank): ${\mathcal{O}}_X^0=0$ and the zero sheaf is locally free of rank $0$; a locally free sheaf is quasi-coherent, its rank is well defined and locally constant, and restriction to an open subscheme preserves local freeness with the same rank.

Geometric vector bundles (Geometric vector bundle with the sections convention): a geometric vector bundle is an affine $X$-scheme with a normalised grading on its relative coordinate algebra, locally graded-isomorphic to $\mathrm{Sym}({\mathcal{O}}_U^r)$; its rank is the rank of the degree-one part ${\mathcal{A}}_1$; the total space of a finite locally free $\mathcal{E}$ is $\mathbb{V}(\mathcal{E})={\mathrm{Spec}}_X\mathrm{Sym}({\mathcal{E}}^{\vee })$, and for $\mathcal{E}=0$ one has $\mathrm{Sym}(0)={\mathcal{O}}_X$ and $\mathbb{V}(0)={\mathrm{Spec}}_X{\mathcal{O}}_X=X$ with the identity structure morphism.

Symmetric algebras (Symmetric algebra of a quasi-coherent module): ${\mathrm{Sym}}^0(\mathcal{F})={\mathcal{O}}_X$, ${\mathrm{Sym}}^1(\mathcal{F})=\mathcal{F}$, and $\mathrm{Sym}(\mathcal{F})$ is generated in degree one, so for $\mathcal{F}=0$ all positive graded parts vanish and $\mathrm{Sym}(0)={\mathcal{O}}_X$; also $\mathrm{Sym}({\mathcal{O}}_X^r)\cong {\mathcal{O}}_X[T_1,\dots ,T_r]$ with degree-one generators, the case $r=0$ being ${\mathcal{O}}_X$ itself.

Relative spectra (Glue relative spectra of affine-local algebras, Affine morphisms are relative spectra): for an affine-locally module-associated algebra sheaf the relative spectrum has ${\pi }^{-1}(U)\cong \mathrm{Spec}\mathrm{\Gamma }(U,\mathcal{A})$ on affine $U\subseteq X$, these charts are compatible under restriction, and the structure morphism is affine with pushforward $\mathcal{A}$; for $\mathcal{A}={\mathcal{O}}_X$ each chart is $\mathrm{Spec}\mathrm{\Gamma }(U,{\mathcal{O}}_X)=U$, so the relative spectrum is $X$ with the identity morphism.

Fibres at a point (Fibre of a module sheaf at a point, The residue field at a point of an affine scheme): the fibre of an ${\mathcal{O}}_X$-module $\mathcal{F}$ at $x$ is the $\kappa (x)$-vector space $\mathcal{F}(x)={\mathcal{F}}_x{\otimes }_{{\mathcal{O}}_{X,x}}\kappa (x)$; for $\mathcal{F}=0$ the stalk and the fibre are $0$, the zero vector space with exactly one element.

Choice accounting: the only Axiom of Choice is the one inherited from the relative-spectrum and symmetric-algebra constructions of [F2] to [F4] (The Axiom of Choice).

Proof technique: direct; compute the local freeness, the symmetric algebra, the relative spectrum and the fibres of the zero module.