TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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Every projective module over a commutative ring is flat

Statement

Every projective module over a commutative ring is flat. This implication requires no form of the Axiom of Choice.

Facts & Assumptions

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

[L1]

A module is flat exactly when tensoring with it preserves injections (Flatness is equivalent to preserving injections and to the ideal and finitely generated ideal tests).

[L2]

Tensor products commute with arbitrary direct sums in either variable; in particular $A \otimes_{R} ⨁_{x \in X} R ≅ ⨁_{x \in X} (A \otimes_{R} R)$ (Tensor products commute with arbitrary direct sums).

[L3]

The regular module is a tensor unit (The regular module is a tensor unit: $R \otimes_{R} N ≅ N$ and $M \otimes_{R} R ≅ M$ ).

[L4]

Every projective module is, without choice, a direct summand of its canonical free cover (Equivalent characterizations of projective modules).

[L5]

Tensor maps preserve identities and compositions (Module homomorphisms induce tensor-product homomorphisms functorially).

Proof

technique · direct

1.1L1L2L3

Every free module $F = ⨁_{x \in X} R$ is flat: for an injection $u : A \to B$ , [L2] and [L3] identify $u \otimes 1_{F}$ with the direct sum of copies of $u$ , which is injective coordinatewise; now apply [L1].

1.2givenL4

By [L4], there are a free module $F$ and homomorphisms $i : P \to F$ , $p : F \to P$ with $p i = {id}_{P}$ .

2.1step 1.2L5

Let $u : A \to B$ be injective and suppose $x \in A \otimes_{R} P$ satisfies $(u \otimes 1_{P}) (x) = 0$ . Functoriality [L5] gives $(u \otimes 1_{F}) ((1_{A} \otimes i) (x)) = (1_{B} \otimes i) ((u \otimes 1_{P}) (x)) = 0$ .

3.1step 1.1step 1.2step 2.1L5

The map $u \otimes 1_{F}$ is injective by step 1.1, so $(1_{A} \otimes i) (x) = 0$ ; applying $1_{A} \otimes p$ and using $(1_{A} \otimes p) (1_{A} \otimes i) = 1_{A} \otimes (p i) = id$ gives $x = 0$ .

4.1step 3.1L1L4∎

Thus $- \otimes_{R} P$ preserves every injection, and [L1] makes $P$ flat. The proof used the canonical splitting supplied by projectivity and made no family of choices.

Depends on

Used by

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. Barr, Acyclic Models, Chapter 2 (standard reference, not scraped)
W. Li, Commutative Algebra, Lectures 9-10 (standard reference, not scraped)