DefinitionDefinition: AI-generatedProof: AI-generated

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Unimodular elements and trivial summands

PDF p. 1: Serre splitting and top Chern obstruction

Definition

Let $A$ be a commutative ring with identity and $P$ a finite projective $A$ -module: $P$ is finitely generated, and maps from $P$ lift through surjective module homomorphisms. An element $p \in P$ is unimodular if there is an $A$ -linear functional $ϕ : P \to A$ with $ϕ (p) = 1$ . A functional $ϕ : P \to A$ is unimodular if it is surjective. A pair $(p, ϕ)$ with $ϕ (p) = 1$ is a unimodular pair.

A vector bundle $E$ on a scheme $X$ is a finite locally free $O_{X}$ -module: every point has an open neighbourhood on which $E$ is a finite direct sum of copies of $O_{X}$ . Its dual is $E^{\lor} = H o m_{O_{X}} (E, O_{X})$ . A global section $s \in Γ (X, E)$ is nowhere zero if, at every point $x$ , its image in the fiber $E \otimes_{O_{X}} k (x)$ is nonzero, where $k (x)$ is the residue field of the local ring $O_{X, x}$ . Equivalently, locally on $X$ the map $O_{X} \to E$ , $1 \mapsto s$ , identifies $O_{X}$ with a direct summand of $E$ . A morphism $E \to O_{X}$ is instead a global section of the dual $E^{\lor}$ .

Assume the Axiom of Choice for the following affine correspondence. On $X = Spec A$ , the bundle associated to a finite projective module $P$ has global sections $P$ ; under this identification a section is nowhere zero exactly when its element is unimodular. A unimodular pair gives an isomorphism $A \oplus \ker ϕ \to P$ , $(a, q) \mapsto a p + q$ . In particular, a surjective functional on $P$ belongs to such a pair.

Justification

The fiber condition uses the actual residue class of the section and is independent of a choice of local frame. The equivalences and the displayed direct-sum decomposition are established below. The affine assertion uses the stated Axiom of Choice through the cited affine module–bundle interface.

Facts & Assumptions

Given: A commutative ring $A$ , a finite projective $A$ -module $P$ , a scheme $X$ with a vector bundle $E$ , and the Axiom of Choice for the affine assertions.

Goal: Justify the local description of nowhere-zero sections and the stated affine equivalences and splitting.

[F1]

Under the Axiom of Choice, finite projective $A$ -modules and vector bundles on $Spec A$ correspond by associated sheaf and global sections, compatibly with localizations and duals; a finite projective module is a direct summand of a finite free module. Affine bundles and finite projective modules

Proof

technique · direct; use local coordinates and the affine module–bundle correspondence

1.1givenconstruct

On a neighbourhood of a point $x$ , choose a frame of $E$ and write $s = (a_{1}, \dots, a_{r})$ . Its fiber value at $x$ is nonzero exactly when some $a_{i}$ has nonzero image in the residue field of $O_{X, x}$ , equivalently is a unit in that local ring. After shrinking the neighbourhood, that coordinate is invertible. Projection to this coordinate followed by multiplication by $a_{i}^{- 1}$ is then a functional taking $s$ to $1$ , so $s$ generates a direct summand. Conversely, if such a local splitting exists, its fiber sends the fiber value of $s$ to $1$ , so that value is nonzero. Thus the two local descriptions agree.

1.2givenalgebraconstruct

If $ϕ (p) = 1$ , the map $(a, q) \mapsto a p + q$ from $A \oplus \ker ϕ$ to $P$ has inverse $v \mapsto (ϕ (v), v - ϕ (v) p)$ . If $ϕ$ is surjective, choose $p$ with $ϕ (p) = 1$ . These statements also hold for the zero ring, where $1 = 0$ .

1.3F1givenconstruct

For the affine converse, write $P$ as a direct summand of $A^{n}$ using [F1]. The resulting coordinate functionals $λ_{i} \in P^{\lor}$ and elements $e_{i} \in P$ satisfy $v = \sum_{i} λ_{i} (v) e_{i}$ for every $v \in P$ . Put $I = (λ_{1} (p), \dots, λ_{n} (p))$ . At a prime $p$ , the fiber of the associated section is zero exactly when all $λ_{i} (p)$ lie in $p$ : one direction follows by applying the functionals and the other from the displayed expansion. If the section is nowhere zero, $I$ is contained in no prime. Under Choice a proper ideal lies in a maximal, hence prime, ideal, so $I = A$ . Coefficients expressing $1$ as a linear combination of the $λ_{i} (p)$ yield a functional $ϕ$ with $ϕ (p) = 1$ .

2.1F1step 1.1step 1.2step 1.3∎

Conversely, if $ϕ (p) = 1$ , its localization and passage to every residue field send the fiber value of $p$ to $1$ ; hence that value is nonzero. Steps 1.1 through 1.3 and this implication establish all the stated equivalences and the splitting.

Depends on

Affine bundles and finite projective modules

Used by · 2

Dependency tree · 2 levels

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