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Local ranks in the affine bundle correspondence

PDF p. 1: Serre splitting theorem

Statement

Assume the Axiom of Choice. If $E$ is a rank- $r$ vector bundle on $X = Spec A$ and $P = Γ (X, E)$ , then $P$ is finite projective and $P_{p}$ is free of rank $r$ over $A_{p}$ for every prime $p$ . A vector bundle is a finite locally free sheaf, and rank $r$ means that its local free rank is $r$ at every point.

Facts & Assumptions

Given: A commutative unital ring $A$ , a rank- $r$ vector bundle $E$ on $Spec A$ , and the Axiom of Choice.

Goal: Identify the projectivity and local ranks of $P = Γ (X, E)$ .

[F1]

The affine bundle–module equivalence preserves rank and identifies the stalk of the associated sheaf at $p$ with the localization of the module. Affine bundles and finite projective modules

Proof

1.1F1given

By the affine equivalence, $P$ is finite projective and its associated sheaf is isomorphic to $E$ .

2.1F1givenstep 1.1∎

At each prime, the stalk of $E$ is free of rank $r$ over the local ring $A_{p}$ by the definition of rank. The stalk identification in [F1] therefore gives $P_{p} ≅ A_{p}^{r}$ .

Depends on

Affine bundles and finite projective modules

Used by · 1

Serre splitting above the dimension

Dependency tree · 3 levels

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