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Projectivity and rank of a free-summand complement

PDF p. 1: Serre splitting theorem

Statement

Assume the Axiom of Choice. Let $P$ be a finite projective module of constant rank $r \geq 1$ over a nonzero commutative ring $A$ . If $P ≅ A \oplus Q$ , then $Q$ is finite projective of constant rank $r - 1$ , and the associated bundles satisfy $\tilde{P} ≅ O_{Spec A} \oplus \tilde{Q}$ . Constant rank means that every prime localization is free of the specified rank.

Facts & Assumptions

Given: $A$ , $P$ , $Q$ , $r$ , the decomposition, and the Axiom of Choice as in the statement.

Goal: Establish finite projectivity, rank, and the associated bundle decomposition.

[F1]

Under Choice, the affine bundle–module correspondence preserves direct sums and rank, and finite projective modules correspond to vector bundles. Affine bundles and finite projective modules

Proof

1.1givenconstruct

The projection of a finite generating set of $P$ generates $Q$ . A direct summand of a projective module is projective: to lift a map from $Q$ through a surjection, precompose it with the projection $P \to Q$ , lift from $P$ , and restrict to $Q$ . Hence $Q$ is finite projective.

2.1F1givenstep 1.1∎

By [F1], $\tilde{P} ≅ O_{X} \oplus \tilde{Q}$ , where $X = Spec A$ , and $\tilde{Q}$ is a vector bundle. At any prime, taking residue-field fibers of this decomposition gives $r = 1 + \dim_{κ (p)} (Q \otimes_{A} κ (p))$ . The rank correspondence in [F1] gives rank $r - 1$ at every prime.

Depends on

Affine bundles and finite projective modules

Used by · 2

Dependency tree · 3 levels

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