Finite locally free sheaves and geometric vector bundles

Setting. The finite locally free ${\mathcal{O}}_X$-modules with ${\mathcal{O}}_X$-linear maps form a category, and a finite locally free ${\mathcal{O}}_X$-module is quasi-coherent with a well-defined locally constant rank by [F2]; the geometric vector bundles over $X$ with the morphisms of [F1] form a category by [F1] and [F13]. Fix the inherited Axiom of Choice as recorded in [F14].

Chart dictionary. Let $(V,\mathcal{A})$ be a geometric vector bundle with structure morphism ${\pi }_V$, let $U=\mathrm{Spec}R\subseteq X$ be affine with a graded isomorphism $\mathcal{A}{\mathrm{\mid }}_U\cong \mathrm{Sym}({\mathcal{O}}_U^r)$ of the bundle data, and let $D(s)\subseteq U$ be a principal open; then ${\pi }_V^{-1}(U)\cong \mathrm{Spec}\mathrm{\Gamma }(U,\mathcal{A})\cong \mathrm{Spec}R[T_1,\dots ,T_r]={\mathbf{A}}_U^r$ as $U$-schemes, the chart of $D(s)$ is the restriction with transition the localization $R[T_1,\dots ,T_r]\to R[T_1,\dots ,T_r{]}_s$, and the degree-one part is ${\mathcal{A}}_1{\mathrm{\mid }}_U\cong {\mathcal{O}}_U^r$. For the total space of a finite locally free $\mathcal{E}$ the same chart reads ${\pi }^{-1}(U)\cong \mathrm{Spec}\mathrm{\Gamma }(U,\mathrm{Sym}({\mathcal{E}}^{\vee }))\cong \mathrm{Spec}R[T_1,\dots ,T_r]$ whenever $\mathcal{E}{\mathrm{\mid }}_U\cong {\mathcal{O}}_U^r$.

The functor $\mathrm{\Phi }$. For a geometric vector bundle $(V,\mathcal{A})$ the degree-one part ${\mathcal{A}}_1$ is finite locally free of rank equal to the rank of the bundle by [F1] and [F2]; put $\mathrm{\Phi }(V,\mathcal{A}):={\mathcal{A}}_1^{\vee }$, finite locally free of the same rank by [F3]. For a morphism $\phi :(V,\mathcal{A})\to (W,\mathcal{B})$ with graded comorphism ${\phi }^{\mathrm{♯}}:\mathcal{B}\to \mathcal{A}$, put $\mathrm{\Phi }(\phi ):=({\phi }_1^{\mathrm{♯}}{)}^{\vee }:{\mathcal{A}}_1^{\vee }\to {\mathcal{B}}_1^{\vee }$, an ${\mathcal{O}}_X$-linear map of finite locally free modules [F3]. Identities and composites are preserved: $\mathrm{\Phi }(\mathrm{i}\mathrm{d})=\mathrm{i}\mathrm{d}$, and for a second morphism $\omega :(W,\mathcal{B})\to (Z,\mathcal{C})$ one has $(\omega \circ \phi {)}_1^{\mathrm{♯}}={\phi }_1^{\mathrm{♯}}\circ {\omega }_1^{\mathrm{♯}}$, whose transpose is $({\omega }_1^{\mathrm{♯}}{)}^{\vee }\circ ({\phi }_1^{\mathrm{♯}}{)}^{\vee }=\mathrm{\Phi }(\omega )\circ \mathrm{\Phi }(\phi )$. Hence $\mathrm{\Phi }$ is a covariant functor $\mathrm{GVB}(X)\to \mathrm{FLF}(X)$ preserving ranks.

Induced morphism from a graded algebra map. Let $\mathcal{A},\mathcal{B}$ be graded ${\mathcal{O}}_X$-algebras that are locally polynomial in the sense that their relative coordinate algebras admit the charts of step 1.2 for the same covering opens; then every graded ${\mathcal{O}}_X$-algebra map $\psi :\mathcal{B}\to \mathcal{A}$ induces an $X$-morphism $\mathrm{\Psi }(\psi ):{\mathrm{Spec}}_X\mathcal{A}\to {\mathrm{Spec}}_X\mathcal{B}$ with comorphism $\psi$. Construction: cover $X$ by the affine opens $U=\mathrm{Spec}R$ over which both algebras are graded free of ranks $s$ and $r$; the global sections map $\psi (U):R[T_1,\dots ,T_s]\to R[T_1,\dots ,T_r]$ is an $R$-algebra map by [F10] under the chart identifications of [F6], giving the $U$-morphism $\mathrm{Spec}(\psi (U))$ between the charts by [F9]. These chart morphisms glue: for a principal open $D(t)\subseteq U$ the restriction of $\psi (U)$ is $\psi (D(t))$ by [F10], so $\mathrm{Spec}(\psi (D(t)))$ is the restriction of $\mathrm{Spec}(\psi (U))$ to the subchart by [F8] and [F9]; hence all chart morphisms agree on overlaps after refining by affine opens, and they glue to an $X$-morphism $\mathrm{\Psi }(\psi )$ because the source is covered by the charts and its structure sheaf is a sheaf [F11]. The comorphism of $\mathrm{\Psi }(\psi )$ restricts to $\psi (U)$ on each chart, whence it equals $\psi$ [F10, F11].

Uniqueness and functoriality of $\mathrm{\Psi }$. If $\phi ,{\phi }^{\mathrm{\prime }}:{\mathrm{Spec}}_X\mathcal{A}\to {\mathrm{Spec}}_X\mathcal{B}$ are $X$-morphisms with the same comorphism $\psi :\mathcal{B}\to \mathcal{A}$, then on each chart $U$ of step 2.1 the restrictions are morphisms of the affine schemes $\mathrm{Spec}\mathrm{\Gamma }(U,\mathcal{A})\to \mathrm{Spec}\mathrm{\Gamma }(U,\mathcal{B})$ with the same global section map $\psi (U)$, hence equal by [F9]; the charts cover the source, so $\phi ={\phi }^{\mathrm{\prime }}$ by [F11]. Therefore the comorphism assignment $\mathrm{\Theta }:\phi \mapsto {\phi }^{\mathrm{♯}}$ is a bijection from the set of grading-preserving $X$-morphisms ${\mathrm{Spec}}_X\mathcal{A}\to {\mathrm{Spec}}_X\mathcal{B}$ onto the graded ${\mathcal{O}}_X$-algebra maps $\mathcal{B}\to \mathcal{A}$, with two-sided inverse $\mathrm{\Psi }$ by step 2.1. Finally $\mathrm{\Psi }(\psi \circ {\psi }^{\mathrm{\prime }})=\mathrm{\Psi }({\psi }^{\mathrm{\prime }})\circ \mathrm{\Psi }(\psi )$ for graded maps $\psi :\mathcal{B}\to \mathcal{A}$ and ${\psi }^{\mathrm{\prime }}:\mathcal{C}\to \mathcal{B}$: both sides are $X$-morphisms ${\mathrm{Spec}}_X\mathcal{A}\to {\mathrm{Spec}}_X\mathcal{C}$ with comorphism $\psi \circ {\psi }^{\mathrm{\prime }}$, and such a morphism is unique; likewise $\mathrm{\Psi }(\mathrm{i}\mathrm{d})=\mathrm{i}\mathrm{d}$.

The functor $\mathbb{V}$. For a morphism $\alpha :\mathcal{E}\to \mathcal{F}$ of finite locally free modules the transpose ${\alpha }^{\vee }:{\mathcal{F}}^{\vee }\to {\mathcal{E}}^{\vee }$ is ${\mathcal{O}}_X$-linear [F3], and $\mathrm{Sym}({\alpha }^{\vee })$ is a graded ${\mathcal{O}}_X$-algebra map by [F4]; define $\mathbb{V}(\alpha ):=\mathrm{\Psi }(\mathrm{Sym}({\alpha }^{\vee })):\mathbb{V}(\mathcal{E})\to \mathbb{V}(\mathcal{F})$, the morphism of step 2.1, which has graded comorphism and hence is a morphism of geometric vector bundles [F1]. Then $\mathbb{V}({\mathrm{i}\mathrm{d}}_{\mathcal{E}})=\mathrm{i}\mathrm{d}$ and $\mathbb{V}(\beta \circ \alpha )=\mathbb{V}(\beta )\circ \mathbb{V}(\alpha )$ for composable $\alpha ,\beta$, because $(\beta \circ \alpha {)}^{\vee }={\alpha }^{\vee }\circ {\beta }^{\vee }$ and Sym preserves composition, and the comorphism determines the morphism by step 3.1. With the object assignment of [F1] this is a covariant functor $\mathrm{FLF}(X)\to \mathrm{GVB}(X)$, and $\mathbb{V}(\mathcal{E})$ has rank $\mathrm{rk}\mathcal{E}$ by [F1] and [F3].

The unit isomorphism. Let $(V,\mathcal{A})$ be a geometric vector bundle. The multiplication map $\mu :\mathrm{Sym}({\mathcal{A}}_1)\to \mathcal{A}$ is a graded ${\mathcal{O}}_X$-algebra map [F4], and it is an isomorphism: on a chart $U$ with $\mathcal{A}{\mathrm{\mid }}_U\cong \mathrm{Sym}({\mathcal{O}}_U^r)$ one has ${\mathcal{A}}_1{\mathrm{\mid }}_U\cong {\mathcal{O}}_U^r$ and $\mu {\mathrm{\mid }}_U$ is the identity of $\mathrm{Sym}({\mathcal{O}}_U^r)$ under this identification [F1, F5], and a morphism of sheaves that is an isomorphism on the members of a cover is an isomorphism [F11]. Under the canonical identification $\mathrm{Sym}(({\mathcal{A}}_1^{\vee }{)}^{\vee })\cong \mathrm{Sym}({\mathcal{A}}_1)$ induced by the evaluation isomorphism $\mathrm{ev}:{\mathcal{A}}_1\to ({\mathcal{A}}_1^{\vee }{)}^{\vee }$ of [F3], the morphism ${\eta }_{(V,\mathcal{A})}:=\mathrm{\Psi }(\mu ):(V,\mathcal{A})\to \mathbb{V}(\mathrm{\Phi }(V,\mathcal{A}))$ is an isomorphism of geometric vector bundles with inverse $\mathrm{\Psi }({\mu }^{-1})$ [step 2.1, F1], and it is natural in $(V,\mathcal{A})$: for a morphism $\phi :(V,\mathcal{A})\to (W,\mathcal{B})$ with comorphism $\psi$, the two composites $\mathbb{V}\mathrm{\Phi }(\phi )\circ {\eta }_V$ and ${\eta }_W\circ \phi$ are $X$-morphisms $(V,\mathcal{A})\to \mathbb{V}(\mathrm{\Phi }(W,\mathcal{B}))$ whose comorphisms $\mathrm{Sym}({\mathcal{B}}_1)\to \mathcal{A}$ are algebra maps agreeing on the degree-one generators, where both send $b\in {\mathcal{B}}_1$ to $\psi (b)={\psi }_1(b)\in {\mathcal{A}}_1$ [F4], so they are equal by the uniqueness of step 3.1.

The counit isomorphism. For a finite locally free $\mathcal{E}$ one has $\mathrm{\Phi }(\mathbb{V}(\mathcal{E}))=((\mathrm{Sym}({\mathcal{E}}^{\vee }){)}_1{)}^{\vee }=({\mathcal{E}}^{\vee }{)}^{\vee }$ by [F4] and step 1.3, and ${\epsilon }_{\mathcal{E}}:={\mathrm{ev}}_{\mathcal{E}}^{-1}:({\mathcal{E}}^{\vee }{)}^{\vee }\to \mathcal{E}$ is an isomorphism of ${\mathcal{O}}_X$-modules by [F3]. It is natural: for $\alpha :\mathcal{E}\to \mathcal{F}$ the identity $({\alpha }^{\vee }{)}^{\vee }\circ {\mathrm{ev}}_{\mathcal{E}}={\mathrm{ev}}_{\mathcal{F}}\circ \alpha$, checked by evaluating functionals on sections, gives ${\epsilon }_{\mathcal{F}}\circ \mathrm{\Phi }\mathbb{V}(\alpha )=\alpha \circ {\epsilon }_{\mathcal{E}}$, where $\mathrm{\Phi }\mathbb{V}(\alpha )=({\alpha }^{\vee }{)}^{\vee }$ by steps 4.1 and 1.3.

The contravariant convention. For finite locally free $\mathcal{F}$ the symmetric algebra $\mathrm{Sym}(\mathcal{F})$ is affine-locally module-associated by [F2] and the affine model of [F4], so $\mathbb{W}(\mathcal{F}):={\mathrm{Spec}}_X\mathrm{Sym}(\mathcal{F})$ is a geometric vector bundle by [F6] and [F7]. The evaluation isomorphism $\mathrm{ev}:\mathcal{F}\to {\mathcal{F}}^{\vee \vee }$ of [F3] induces $\mathrm{Sym}(\mathcal{F})\cong \mathrm{Sym}({\mathcal{F}}^{\vee \vee })=\mathrm{Sym}(({\mathcal{F}}^{\vee }{)}^{\vee })$, and relative spectra of isomorphic locally polynomial graded algebras are canonically isomorphic by step 3.1 applied to the algebra isomorphism and its inverse; hence $\mathbb{W}(\mathcal{F})\cong \mathbb{V}({\mathcal{F}}^{\vee })$, and a morphism $\alpha :\mathcal{F}\to \mathcal{G}$ induces $\mathbb{W}(\mathcal{G})\to \mathbb{W}(\mathcal{F})$ as in step 4.1, so $\mathbb{W}$ is the contravariant coordinate-module convention. Its quasi-inverse is given by the degree-one coordinate part: for a geometric vector bundle $(V,\mathcal{A})$ the multiplication isomorphism of step 4.2 identifies $(V,\mathcal{A})$ with ${\mathrm{Spec}}_X\mathrm{Sym}({\mathcal{A}}_1)=\mathbb{W}({\mathcal{A}}_1)$, so $\mathcal{F}={\mathrm{Sym}}^1$ is recovered [F4, step 4.2].

Equivalence. By steps 4.2 and 5.1 the functors $\mathrm{\Phi }$ and $\mathbb{V}$ are quasi-inverse: $\eta :1_{\mathrm{GVB}(X)}\Rightarrow \mathbb{V}\mathrm{\Phi }$ of step 4.2 and $\epsilon :\mathrm{\Phi }\mathbb{V}\Rightarrow 1_{\mathrm{FLF}(X)}$ of step 5.1 are natural isomorphisms. Hence $\mathrm{\Phi }:\mathrm{GVB}(X)\to \mathrm{FLF}(X)$ is an equivalence of categories with quasi-inverse $\mathbb{V}$, equivalently $\mathbb{V}$ is an equivalence with quasi-inverse the sheaf of sections $\mathrm{\Phi }$ [F13]; ranks correspond by steps 1.2, 4.1 and 1.3.

Full faithfulness. For finite locally free $\mathcal{E},\mathcal{F}$ the comorphism bijection of step 3.1, applied to $\mathcal{A}=\mathrm{Sym}({\mathcal{E}}^{\vee })$ and $\mathcal{B}=\mathrm{Sym}({\mathcal{F}}^{\vee })$, identifies morphisms $\mathbb{V}(\mathcal{E})\to \mathbb{V}(\mathcal{F})$ of geometric vector bundles with graded algebra maps $\mathrm{Sym}({\mathcal{F}}^{\vee })\to \mathrm{Sym}({\mathcal{E}}^{\vee })$, which by the universal property of [F4] correspond bijectively to ${\mathcal{O}}_X$-linear maps ${\mathcal{F}}^{\vee }\to {\mathcal{E}}^{\vee }=(\mathrm{Sym}({\mathcal{E}}^{\vee }){)}_1$, that is, by the double dual isomorphism of [F3], to ${\mathcal{O}}_X$-linear maps $\mathcal{E}\to \mathcal{F}$; the resulting bijection sends $\alpha$ to $\mathbb{V}(\alpha )$ by step 4.1, with inverse $\phi \mapsto {\epsilon }_{\mathcal{F}}\circ ({\phi }_1^{\mathrm{♯}}{)}^{\vee }\circ {\mathrm{ev}}_{\mathcal{E}}$. Indeed for $\alpha$ the composite is $\alpha$ by the naturality identity of step 5.1, and for $\phi$ the induced map $\alpha$ satisfies ${\alpha }^{\vee }={\phi }_1^{\mathrm{♯}}$ because double transposition returns the original map, so $\mathrm{Sym}({\alpha }^{\vee })={\phi }^{\mathrm{♯}}$ since a graded algebra map out of $\mathrm{Sym}({\mathcal{F}}^{\vee })$ is determined by its degree-one part, and hence $\mathbb{V}(\alpha )=\phi$ by step 3.1.

Choice accounting. The Axiom of Choice is used only through the associated-sheaf, gluing, symmetric-algebra and relative-spectrum constructions inherited in [F1] and [F4] to [F7], as recorded in [F14]; the constructions of this proof range over the families of all affine charts on which the algebras are graded free and of all module morphisms, and select no simultaneous family of charts, generators or isomorphisms. The rank of each side is determined independently of charts by [F2], so the equivalence and the rank correspondence are stated in the inherited AC theory.