LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite projective modules are finitely presented

PDF p. 1: Serre splitting theorem

Statement

Let $A$ be a commutative unital ring and $P$ a finite projective $A$ -module. Then $P$ is finitely presented: there is an exact sequence $A^{m} \to A^{n} \to P \to 0$ with $m, n$ finite. Here finite projective means finitely generated and having the lifting property through surjective module maps.

Facts & Assumptions

Given: A commutative unital ring $A$ and a finite projective $A$ -module $P$ .

Goal: Construct a finite presentation of $P$ .

Proof

1.1givenchooseconstruct

Choose finitely many generators of $P$ and let $q : A^{n} \to P$ be the resulting surjection. Projectivity lifts the identity of $P$ through $q$ , giving a section $s : P \to A^{n}$ . Thus $A^{n} = s (P) \oplus \ker q$ .

2.1step 1.1construct∎

The projection $A^{n} \to \ker q$ maps the standard basis to a finite generating set of $\ker q$ . Mapping a finite free module onto that kernel and composing with its inclusion in $A^{n}$ gives the required presentation.

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Serre splitting above the dimension