Affine bundles and finite projective modules

First suppose $X$ is empty. Under AC a nonzero commutative ring has a prime ideal, so $A=0$. The only $A$-module and the only sheaf of modules on $X$ are zero; all assertions, including the empty rank function, follow. Hence assume $X$ is nonempty below.

We record a finite-generation patching argument. If an $A$-module $M$ has $M_{f_i}$ finitely generated for a finite principal cover, choose finitely many elements of $M$ whose images generate each $M_{f_i}$: clear the denominators of local generators. Let $N$ be the submodule they generate. For any $m\in M$, each $m\mathrm{/}1$ lies in $N_{f_i}$, so some power $f_i^{n_i}m$ lies in $N$. By [F2] suitable coefficients give $1={\sum }_i{a}_i{f}_i^{n_i}$, whence $m={\sum }_i{a}_i{f}_i^{n_i}m\in N$. Thus $M$ is finitely generated.

If, in addition, each $M_{f_i}$ is finite free, choose a surjection $q:A^n\to M$ using step 2.1 and put $K=\ker q$. Localization is exact: fractions representing a relation become actual relations after multiplication by a denominator, and the same argument detects zero fractions. Hence $K_{f_i}=\ker (q_{f_i})$. Since the free module $M_{f_i}$ admits lifts of its finitely many basis vectors through $q_{f_i}$, this localized surjection splits; consequently $K_{f_i}$ is a direct summand of $A_{f_i}^n$ and is finitely generated by the projections of the standard basis. Step 2.1 applied to $K$ shows that $K$ is finitely generated. Thus $M$ is finitely presented: there is an exact sequence $A^m\to A^n\stackrel{q}{\to }M\to 0$ with $m,n$ finite.

A finite presentation also permits clearing denominators of maps: for any $A$-module $N$ and $f\in A$, every $A_f$-linear map $M_f\to N_f$ is $f^{-k}{s}_f$ for some $k\ge 0$ and $A$-linear map $s:M\to N$. Indeed choose images in $N$ for the finitely many generators after clearing a common denominator; the finitely many relations vanish after localization, so multiplying all chosen images by one further power of $f$ makes every relation vanish in $N$. The resulting assignment defines $s$. Moreover, if a map from the finitely generated module $M$ becomes zero after localization at $f$, a single power of $f$ annihilates that map: take a common power killing the images of its finite generating set.

For $M$ as in step 3.1, the localized surjection $q_{f_i}:A_{f_i}^n\to M_{f_i}$ has a section $t_i$. By step 4.1, after multiplying $t_i$ by a power of $f_i$ it is the localization of an $A$-linear map $s_i:M\to A^n$. Multiplying by a further power if necessary gives $q{s}_i=f_i^{N_i}{\mathrm{id}}_M$ as maps over $A$. Since the powers $f_i^{N_i}$ generate the unit ideal, choose $a_i$ with ${\sum }_i{a}_i{f}_i^{N_i}=1$. Then $s={\sum }_i{a}_i{s}_i$ satisfies $qs={\mathrm{id}}_M$. Thus $M$ is a direct summand of $A^n$, and is finite projective by [F4].

Let $E$ be finite locally free. By [F1], $E\cong \tilde{M}$ for $M=\mathrm{\Gamma }(X,E)$. Refine its local free charts by distinguished opens and use compactness to obtain a finite cover $D(f_i)$ on which $E\cong {\mathcal{O}}^{r_i}$. By [F3], $M_{f_i}\cong A_{f_i}^{r_i}$. Steps 3.1 and 5.1 show that $M$ is finite projective, including the finite-presentation assertion needed for the restriction of the equivalence.

Conversely, let $P$ be finite projective. A finite generating surjection $A^n\to P$ splits by its lifting property, so $P$ is the image of an idempotent matrix $e$ on $A^n$. At a prime $\mathfrak{p}$, the reduction of $e$ over the residue field is an idempotent and therefore has a basis in which it is $\mathrm{diag}(I_r,0)$. Lift that change-of-basis matrix to $A_{\mathfrak{p}}$; its determinant is a unit there, hence it is invertible. After this change of basis, the upper-left $r\times r$ block of $e$ is invertible at $\mathfrak{p}$. Shrinking to a principal neighbourhood of $\mathfrak{p}$ makes both matrices invertible. Elementary block row and column operations then identify the image of $e$ with the direct sum of a free rank-$r$ module and the image of an idempotent matrix $d$ whose entries vanish modulo $\mathfrak{p}$. More explicitly, the image and kernel of $e$ split the ambient module; the invertible upper-left block gives a rank-$r$ free summand of the image and reduces the remaining idempotent to zero modulo $\mathfrak{p}$. The determinant of $I-d$ is therefore outside $\mathfrak{p}$; after a further principal shrinking it is invertible. Since $d(I-d)=0$, this forces $d=0$ there. Hence $P_f\cong A_f^r$ on a principal neighbourhood of each prime. By [F3], $\tilde{P}$ is finite locally free.

Steps 6.1 and 7.1 show that the quasi-inverse functors of [F1] take precisely the stated subcategories into one another, so their unit and counit restrict to a quasi-inverse equivalence. The localization $P_{\mathfrak{p}}$ is free by step 7.1, and [F3] identifies it with the stalk of $\tilde{P}$; therefore its rank agrees with the bundle rank in the published definition. This also makes the module rank well-defined.

For finite projective $P,Q$, the canonical sheaf maps $\tilde{P}\oplus \tilde{Q}\to \widetilde{P\oplus Q}$, $\tilde{P}\otimes \tilde{Q}\to \widetilde{P{\otimes }_{A}Q}$, and $\widetilde{P^{\vee }}\to \mathcal{H}om(\tilde{P},{\mathcal{O}}_X)$ are isomorphisms. Indeed, use a common principal cover on which $P$ and $Q$ are finite free; on each member these maps are the standard finite-free identifications, including ${\mathrm{Hom}}_{A_f}(A_f^r,A_f)\cong A_f^r$. For the dual, the identification $(P^{\vee }{)}_f\cong {\mathrm{Hom}}_{A_f}(P_f,A_f)$ follows either from the finite presentation in step 3.1 and clearing denominators as in step 4.1, or from the finite free-summand representation of $P$. Isomorphism of sheaf maps is local. The inverse equivalence consequently preserves the same operations up to their canonical isomorphisms.

Let $0\to E^{\mathrm{\prime }}\to E\to E^{\mathrm{\prime }\mathrm{\prime }}\to 0$ be a short exact sequence of vector bundles on $X$. By [F1] write it as associated sheaves of module maps $P^{\mathrm{\prime }}\to P\stackrel{q}{\to }{P}^{\mathrm{\prime }\mathrm{\prime }}$. Localization identifies their stalks with the localizations of these maps by [F3]. Exactness at all stalks implies exactness of $0\to P^{\mathrm{\prime }}\to P\to P^{\mathrm{\prime }\mathrm{\prime }}\to 0$: a module whose localization at every prime is zero is zero, since a nonzero element has a proper annihilator contained in a prime. Apply this to the kernel, homology and cokernel, using exactness of localization. By step 6.1, $P^{\mathrm{\prime }\mathrm{\prime }}$ is projective; [F4] gives a section of $q$. Its associated sheaf is a section of $E\to E^{\mathrm{\prime }\mathrm{\prime }}$, and the original sequence splits.

Steps 8.1, 9.1 and 10.1 prove all assertions. AC is used through the cited affine equivalence, compactness and prime-ideal consequences, and the cited projective-module characterization.